Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UF00097847/00001
## Material Information- Title:
- On Borsuk's paste job and related topics
- Added title page title:
- Borsuk's paste job
- Creator:
- Borrego, Joseph Thomas, 1939-
- Publication Date:
- 1966
- Copyright Date:
- 1966
- Language:
- English
- Physical Description:
- iii, 100 leaves : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Cardinality ( jstor )
Continuous functions ( jstor ) Hausdorff spaces ( jstor ) Homomorphisms ( jstor ) Isomorphism ( jstor ) Mathematical congruence ( jstor ) Mathematics ( jstor ) Paste ( jstor ) Semigroups ( jstor ) Sufficient conditions ( jstor ) Dissertations, Academic -- Mathematics -- UF Mathematics thesis Ph. D Topology ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis - University of Florida.
- Bibliography:
- Bibliography: leaves 98-99.
- Additional Physical Form:
- Also available on World Wide Web
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000554478 ( AlephBibNum )
13406462 ( OCLC ) ACX9321 ( NOTIS )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

ON BORSUK'S PASTE JOB AND RELATED TOPICS By JOSEPH THOMAS BORREGO, JR. A DISSERTATION PRLSENThED TO THIE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DFGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA April, 1966 ACKNOWLEDGEMENTS The author wishes to thank his advisor, Professor A. D. Wallace, for the time, encouragement, and advice given to the author in the preparation of this thesis. The author also wishes to thank Professor Wallace for his aid in the professional development of the author. The author wishes to thank Professor G. A. Jensen for her aid in proof reading the manuscript and for her many valuable suggestions. The author wishes to thank Professor J. M. Day for many interesting conversations and acknowledges the fact that Theorem 3.6.3 was a direct result of one of these conversations. The author wishes to thank the other members of his committee, Professors A. R. Bednarek, D. J. Foulis, S. P. Franklin, T. O. Moore, A. R. Quinton, and F. M. Sioson. The author wishes to thank Professor W. L. Strother for being a willing listener. The author wishes to thank Professor J. E. Maxfield, Chairman of the Department of Mathematics, for many kind- nesses. The author wishes to thank Mrs. Juanita Patterson for her care in typing the manuscript. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS . . . . . . . . .. ii INTRODUCTION . . . . . . . . ... 1 Chapter I. PRELIMINARY PROPOSITIONS AND DEFINITIONS 3 II. HOMOTOPIC BORSUK'S PASTE JOBS HAVE THE SAME COHOMOLOGICAL STRUCTURE . . .. 17 III. BORSUK'S PASTE JOB IN SEMIGROUPS ... . 36 IV. HOMOMORPHIC RETRACTS IN SEMIGROUPS . 70 V. APPLICATION OF HOMOMORPHIC RETRACTIONS TO RELATIVE IDEALS ............ 85 BIBLIOGRAPHY . . . . . . . . ... . 98 BIOGRAPHICAL SKETCH . . . . . . . .. .100 iii INTRODUCTION The idea of Borsuk's Paste Job was first introduced in 1935 by Borsuk [1] who called it a singular retract. Other authors [3], [6] have referred to Borsuk's Paste Job as an adjunction space. At the present time, rela- tively little is known about the structure of these spaces; however, Chapters II and III of this thesis contain some new information. Chapter IV and V deal with the study and application of an idea which arises in Chapter III. Chapter I is an introductory chapter, part of which contains preliminary definitions and propositions which are needed in the other chapters. The remainder of the chapter is devoted to some propositions which are developed in or- der to deal with certain examples. In Chapter II it is proved that if two Borsuk's Paste Jobs have homotopic defining maps then the two Bor- suk's Paste Jobs have isomorphic cohomology groups. By using a generalization of the definition of homotopic maps, a generalization of this result is also obtained. If the construction of Borsuk's Paste Job is done on a topologicall) semigroup, then does the resulting Borsuk's Paste Job admit a natural semigroup structure is the question which is asked in Chapter III. The chapter studies a partic- ular aspect of this question, namely, the determination of necessary or sufficient conditions for an affirmative an- swer to the above question for all possible defining homo- morphisms on a fixed subsemigroup. Section 1 contains some necessary conditions and an indication of why attention has been restricted to homomorphisms and subsemigroups; Section 2 gives several sufficient conditions of the above type; Section 3 contains some examples and counter-examples; Sec- tion 4 gives necessary and sufficient conditions for an af- firmative answer under the hypothesis that the subsemigroup consists of left zeros; Section 5 extends the results of Section 4 to cover the case of minimal ideals; and Section 6 is an investigation of the connection of the question with congruences. The idea of a homomorphic retract, which is intro- duced in Chapter III, is studied further in Chapter IV. Sufficient conditions for various types of subsemigroups to be homomorphic retracts are given. In particular, two sets of necessary and sufficient conditions for the minimal ideal to be a homomorphic retract are presented. In Chapter V, under the hypothesis that T is a homo- morphic retract, the retraction properties of minimal T- ideals [17], [18], [19] are studied. Sufficient conditions are given to insure that the minimal T-ideals are retracts. CHAPTER I PRELIMINARY PROPOSITIONS AND DEFINITIONS 0: Quotient Spaces. A relation on a set X is a subset of X x X. If R is a relation on X and if A is a subset of X, then the fol- lowing notation is used: AR = (yl(x,y) e R for some x e A], R(-1) = ((y,x)l(x,y) e R], RA = AR(- ), RR = ((x,z)l(x,y) and (y,z) e R for some y e X]. Let R be a relation on X, then R is reflexive in case AX = ((x,x)Ix e X] is a subset of R, R is symmetric if and only if R(-1) = R, and R is transitive whenever RoR is a subset of R. An equivalence relation is a reflexive, symmetric and transitive relation. If R is an equivalence relation on X, then [xRlx e X) is called the factor set of X by R and is denoted by X/R. The function cp from X to X/R defined by cp(x) = xR is called the natural projection of X onto X/R. If X is a topological space and R is an equivalence relation on X, then X/R is topologized by the smallest to- pology for which the natural projection is continuous. More explicitly, a subset U of X/R is open if and only if -p (U) is open. This topology on X/R is called the quotient topol- oqy [7] and it is assumed, without further mention, that all factor sets of topological spaces are endowed with the quotient topology. 1.0.1: Lemma. If R is an equivalence relation on a to- pological space X and if f is a function whose domain is X/R, then f is continuous if and only if fcp is continuous. Proof: See [7, p. 96J. If X is a Hausdorff space and R is an equivalence relation which is defined on X, then X/R need not be Haus- dorff [5, p. 132]. In the next lemma conditions are given which insure that X/R is Hausdorff. 1.0.2: Lemma. If X is a compact Hausdorff space and R is a closed equivalence relation on X, then X/R is Hausdorff. This is a direct consequence of [12, Theorem 9] and [5, p. 13]. 1: Definition of Borsuk's Paste Job. The type of construction described here was first given by Karol Borsuk [1] and was called a singular retract. Throughout this section, X and Y are compact Haus- dorff spaces, A is a closed subspace of X, and f is a con- tinuous function from A onto Y. 1.1.1: Definition. (a) If f x f is the function from A x A to Y x Y defined by (f x f)(a,b) = (f(a),f(b)), then -l R(f,X) = [(f x f) (A Y)] U A X. It is easy to see that R(f,X) is a closed equivalence relation on X and that (x,y) R(f,X) if and only if f(x) = f(y) or x = y. (b) Let Z(f,X) = X/R(f,X). Then Z(f,X) is called Borsuk's Paste Job of Y to X by f. It follows from Lemma 1.0.2 and 11.1. (a) that Z(f,X) is a compact Hausdorff space. The following lemma is easily proved using 1.1.1. 1.1.2: Lemma. If p is the natural projection of X onto Z(f,X), then there exists a function k from Y into Z(f,X) such that kf = pi, where i is the injection of A into X, i.e., the following diagram is analytic. X > Z(f,x) i k A Y Moreover, k is a homeomorphism of Y onto O(A) and 0 maps (X--A) homeomorphically onto (Z(f,X)-k(Y)). This lemma says that Z(f,X) = J(X--A) U k(Y) and that c(X--A) is topologically the same as (X--A) and k(Y) is topologically the same as Y. This does not imply that the structure of Z(f,X) is completely determined by Lemma 1.1.2, for let X be the unit disk E2 and let A = Y be the boundary of E2, i.e., S If f(z) = z for all z e S and 2 1 2 f2(z) = z for all z C S then Z(fl,E ) is topologically E2 and Z(f2,E ) is the projective plane. 2: Preliminary Propositions on the Cohomoloqical Structure of Borsuk's Paste Job. All of the cohomology groups in this thesis are the Alexander-Kolmogoroff Cohomology groups [11], [14]. 1.2.1: Lemma. Let X and Y be topological spaces, let A and B be closed subsets of X and Y, respectively, let f be a continuous function from (X,A) to (Y,B), let the continu- ous function g from X to Y be defined by g(x) = f(x), and let the continuous function h from A to B be defined by h(x) = f(x). If any two of the three induced homomorphisms f*, g*, and h* are isomorphisms, then the third is also an isomorphism [4]. Proof: Suppose f* and g* are isomorphisms. In the follow- ing analytic diagram, j, j', i, and i' are inclusion map- pings, 6 and 6' are co-boundary operators, and the rows are exact [14]. j* i* 6 j* HP(B HY H ) H (Y,B)--- H( H(B)' H)-- H-(y) f* g* h* f* g* v V v V V j* i'* 6 j '* HP(X,A)--- HP(X)----- > HP(A)--- H (X,A)--- HP- (X) Since f* and g* are isomorphisms, the Five Lemma [9] says that h* is also an isomorphism. The other two cases are proven in a similar fashion, provided that HP(Z,K) is interpreted as f0} whenever p < 0. In the remainder of this chapter, X and Y are compact Hausdorff spaces, A is a closed subset of X, f is a continu- ous function from A onto Y, W is the natural projection of X onto Z(f,X), and k is the homeomorphism given by Lemma 1.1.2. With this notation, the following diagram is analy- tic. cp X zZ(f,X) U k A >Y 1.2.2: Lemma. The equation p(x) = q(x) defines a con- tinuous function 0' from (X,A) onto (Z(f,X), k(Y)). The induced homomorphism p'* is an isomorphism. The lemma follows immediately from Lemma 1.1.2 and the Map Excision Theorem [13]. This lemma says that for the proper choice of homo- morphisms there is an exact sequence: H (X,A)-> H (Z(f,X)) -- H (Y)-> H1(X,A)--- ... --> HP-(Y)---> HP(X,A)-> HP(Z(f,X))--> HP(Y)--- HP+1(X,A)--- .... 1.2.3: Lemma. The induced homomorphism cp* is an isomor- phism if and only if f* is an isomorphism. Proof: Let cp' be as in Lemma 1.2.2 and cp" be the continu- ous function from A to k(Y) which is defined by cp"(x) = p(x) for all x e A. By Lemma 1.1.2 it follows that cp"* = (kf)* = f*k*. Thus cp"* is an isomorphism if and only if f* is an isomorphism. According to Lemma 1.2.1, c"* is an isomor- phism if and only if c* is an isomorphism, and so p* is an isomorphism if and only if f* is an isomorphism. 1.2.4: Lemma. k(Y) is a retract of Z(f,X) if and only if f is extendable to X. Proof: See [6, p. 10]. 1.2.5: Corollary. If f is extendable to X, then HP(Z) is isomorphic to HP(X,A) x HP(Y). Proposition 1.2.9 provides a method for applying the Absolute Mayer-Vietorsis Sequence [14] to Borsuk's Paste Job. First, a few lemmas which are needed in Pro- position 1.2.9 are proved. 1.2.6: Lemma. Let B be a closed subset of X, let A' = B n A and let f' be the restriction of f to A'. Then Z(f',B) can be topologically embedded in Z(f,X). Proof: Define h from Z(f',B) into Z(f,X) by h(z) e i', (z) for all z C Z(f',B) where 0' is the natural pro- jection of B onto Z(f',B). It is enough to show that h is a homeomorphism. To see that h is well defined, let x,y e B such that ('(x) = 0'(y). By 1.1.1, x = y or f'(x) = f'(y) so that x = y or f(x) = f(y) and c(x) = e(y). Lemma 1.0.1 implies h is continuous since hc' is cp restricted to B. Thus it remains only to show that h is one-to-one. Let x,y e B such that he'(x) = he'(y). Then t(x) = he'(x) h,'(y) = e(y) so that x = y or f(x) = f(y). It follows that (x) = '(y). 1,2.7: Lemma. Let B1 and B2 be closed subsets of X, let A. = B. n A, let f. be the restriction of f to Ai, and let h. be defined as in Lemma 1.2.6, where i = 1 or 2. If 1 B1 U B2 = X, then hl(Z(fl,B1) U h2(Z(f2,B2))= Z(f,X). Proof: If z e Z(f,X), then there exists x c X such that p(x) = z. But x e B1 or x e E2 so that z = hi.pi(x) where .i is the natural projection of B. onto Z(fi,B.) and i = 1 or 2. 1.2.8: Lemma. Let B1 and B2 be closed subsets of S, let B = B1 B, let A = B. A, let f. be the restriction of f to Ai, and let h. be as defined in Lemma 1.2.6, where i = 1, 2, or 3. If f(A3) = f(A1) n f(A2), then h3(Z(f3,B3)) = h (Z(fl,B1)) n h2(Z(f2,B2)). Proof: If h3(Z) e h3(Z(f3,B3)), then there exists a point x in B3 such that cp3(x) = z. Thus h3(z) e h (Z(fl,B1)) n h2(Z(f2,B2)) since h3(z) = p(x) = hlcpl(x) = h222(x). If z e hl(Z(f,,B1)) n h2(Z(f2,B2)), then there exist xl e BL and x2 e B2 such that cp(x) = z = p(x2). Thus xI = x2, or f(xl) = f(x2). If xl = x2, then x, e B3 and h3p3(x1) = c(xl) = z. If f(x1) = f(x2), then there exists x3 E A3 such that f(x3) = f(x2) = f(x ). Hence h33 (x3) = p(x3) = p(x1) = z. 1.2.9: Proposition. Let B1 and B2 be closed subsets of X, let B = B1 B, A = B i i n A, and let f. be the re- striction of f to A If Bl U B2 = X and f(A3) = f(A1) n f(A2), then the following sequence with the obvious choice of homomorphisms is exact. H (Z(f19B1)) H (Z(f,X)) > > H (Z(f3,B3)) H (Z(f2,B2)) Hl(z(f,X)) > ... > HP- (Z(f3,B3) HP(z(fIB1) HP(z(f,X)) -- > X HP(Z(f3, B))--> H(z(f2,B2)) Hp+l(z(f,X)) > ... This proposition is an immediate consequence of Lemmas 1.2.7 and 1.2.8 and the statement of the Absolute deyer-Vietorsis Sequence. 2 1 2 If E is the unit disk, if S is the boundary of E and if f is a continuous function from S onto S then HP(Z(f,E2)) is easily computed for each positive integer p. This computation will follow from the next two lemmas. Notation: If A and B are groups and if g is a homo- morphism from A into B, then K[g] denotes the kernel of g and I[g] denotes the image of g. The continuous function p' from (X,A) to (Z(f,X),k(y)) is defined by cd(x) = cp(x). rhis notation will be used in the remainder of this section. 1.2.10: Lemma. (i) If HP(Y) = 0 and p > 0, then P(Z(f,X)) is isomorphic to HP(X,A)/6f* HP-(Y), where 5 is the co-boundary operator from H P-(A) to HP(X,A). (ii) If H (y) = 0, then H (Z(f,X)) is isomorphic to H (X,A). Proof: In the following analytic diagram i and i' are in- jections, 6 and 6' are co-boundary operators. H-l(k(Y))-- HP(Z(f,X),k(Y))-- HP(Z(f,X))--> 0 k* HP-l(y) cp'* cp f* 6 i* HP-1(A) HP(X,A) > HP(X) Since the fact that the rows of the diagram are exact implies i'* is onto, HP(Z(f,X)) is isomorphic to HP(Z(f,X), k(Y))/K[i*]. From the analyticity of the diagram and the fact that k* is an isomorphism, it follows that K[i*] = 6'[HP-lk(Y)] = 1'*-6f*HP-1(Y). Since cp'* is an isomor- phism, HP(Z(f,X),k(Y))/K[i*] is isomorphic to HP(X,A)/ 6f*'[HP-1( ) . The exactness of the sequence 0 -- H (X,A) -> H (Z(f,X)) -> 0 implies (ii). Let En be the n-cell and let S n-be the boundary of En. If the coefficient group is the group of integers G, then Hn-1(S"n-) is the group of integers. If f is a continuous function from Sn- to Sn-, then deg f = f*(l) [6]. Thus, from the lemma and the fact that 6 is an iso- morphism if X = En and A = Y = Sn-1, it follows that Hn(Z(f,E )) is isomorphic to H (E,Sn-l)/f*H (S ), which is also isomorphic to G/(deg f)G. 1.2.11: Lemma. If HP-1(X,A) = 0, then HP-1(Z(f,X)) is isomorphic to K[6f*] where 6 is the co-boundary operator from HP-1(A) to HP(X,A). Proof: In the following analytic diagram j is an inclusion map and the top row is exact. j 6' 0 ---> HP-l(Z(f,X)) --> HP-(k(Y)) -- HP(Z(f,X),k(Y)) 1 k* HP-1y) c'* I f* HP- (A) > HP(X,A) Thus j* is a monomorphism and H -(Z(f,X)) is isomorphic to K[6'] = K[c'*-16f*k*], which is isomorphic to K[6f*] since cp* and k* are isomorphisms. If X = En and Y = Sn-1 = A, then Hn-1(Z(f,En)) is isomorphic to K[f*] = [(gg e G and (deg f)g = 0) since 6 is an isomorphism. It follows that if f is a continuous function from S n-onto S n-, then H (f,En) is isomorphic to G, [gl(deg f)g = 0), G/(deg f)G, or 0, according as p = 0, p = n-1, p = n, or p 0, n-1, n. The following corollary is clear because of these computations. 1.2.12: Corollary. If f and g are continuous maps from S n-onto S n-and the coefficient groups are the group of integers, then Hn(Z(f,En)) is isomorphic to Hn(Z(g,En))if and only if deg f = deg g. 3i Elementary Propositions on the Point-Set Structure of Z(f,X). Let g be a continuous function from Y onto another compact Hausdorff space T. The first lemma in this section gives a relationship between Z(gf,X) and Z(gk- Z(f,X)). 1.3.1: Lemma. Z(gf,X) is homeomorphic to Z(gk -, Z(f,X)). 1.3.1: Lemma. Z(gf,X) is homeomorphic to Z(gk Z(f,X)). 13 Proof: Let cp' and cp" be natural projections with ranges and domains as indicated below: X Z(gf,X) U k gf A T fp -1 X > Z(f,X) > Z(gk ,Z(f,X)) U U k' kf gk A k(Y) T -i It will be shown that the equation h(z) = c' cp c" (z) de- -l fines a homeomorphism from Z(gf,X) onto Z(gk Z(f,X)). To see that h is well defined, let x, y e X such that p"(x) = cp"(y). This implies that x = y or gf(x) = gf(y). Thus if x = y, then it is clear that p'cp(x) = (p'cp(y), and if gf(x) = gf(y), then k'gk-lkf(x) = k'gk-lkf(y). -i But k'gk = cp' and kf = p so that cp'cp(x) = c'cp (y). It is clear that h is continuous since hcp" = cp'cp and the latter is continuous. The fact that h is onto is an immediate consequence of the fact that if z c Z(gk ,Z(f,X)), then there exists a point x c X such that z = p'cp (x) = h cp"(x). It remains only to show that h is one-to-one. If x,y c X such that p'p(x) = p'cp(y), then x = y or k"gk -kf(x) = k'gk- kf(y). If x = y, then p"(x) = p"(y). If k'gk-lkf(x) = k"gk kf(y), then cp"(x) = gf(x) = gf(y) = cp"(y) since k' is one-to-one. This lemma gives a way to break up the study of Z(f,X) into two cases: (1) f is monotone and (2) f is light. It is known that a continuous function f may be re- presented as the composition of a monotone continuous func- tion g and a light continuous function g'. Thus, Z(f,X) is homeomorphic to Z(g',Z(g,X)) and Borsuk's Paste Job can be divided into a light part and into a monotone part. The difficulty is that even if A is a "very nice space", then g(A) need not be well-behaved and Z(g,X) can be even more pathological (See [5]). However, in case X is the two-cell, E2 and A is the one-sphere S then things are not too bad. 1.3.2: Remark. If g is a continuous monotone function de- fined on S then Z(g,E2) is the two-cell or two-sphere. Proof: Since g is monotone and continuous, g(S ) is a point, or S1 and the natural projection ep' of E2 onto Z(g,E2) is monotone [21]. If the continuous monotone image of E2 is cyclic, then the image is a two-cell or a two- sphere [21, p. 173]. Thus, it is sufficient to show Z(g,E2) 2 is cyclic. Suppose Z(g,E ) is not cyclic, i.e., there ex- ists z e Z(g,E2) such that z separates Z(g,E ), then there exist open disjoint sets K and T such that K U T = Z(g,E2 ) (z]. The sets p' 1(T) and c'1 (K) are open disjoint sets and E2 -(z) = cp (T) U '(K). If p' (z) is con- 2 -1 trained in the interior of E2 then cp' (z) is a single point and since single points do not separate E it fol- lows that c (z) c S However, subsets of S do not sep- 2 2 arate E2 and therefore Z(g,E ) must be cyclic. If Z(g,E2) is a two-sphere, then the same theorem just quoted says cp'(S ) is a point and so Y must be a 2 point. Since this case is trivial, it can be said Z(g,E ) 1 1 is a two-cell and g(S ) is S Thus, it is enough to study the light maps defined on S in order to determine 2 the structure of Z(g,E ). 1.3.3: Corollary. If f is a continuous monotone function from S into S then deg f = 0, 1, or -1. Proof: If f is not onto, then deg f = 0. The case of 2 interest is when f is onto. If Z(f,E ) is a two-cell, then Z(f,E2 ) has the cohomology groups of a point and from the computation in Section 2 it is seen that deg f = 1. The next question considered is: If f' is a con- tinuous function from A onto a compact Hausdorff space Y', how "similar" does Y' have to be to Y to insure HP(Z(f,X)) is isomorphic to HP(Z(f',X)). Let X = E2 and A = Y = Y' = S', then HP(Z(f,X)) is isomorphic to HP(Z(f',X)) if and only if deg f = + deg f'. Thus, it is seen that in addi- tion to restrictions on Y and Y' some restrictions must be placed on f and f'. 1.3.4: Remark. If g is a continuous function from Y to Y', if g* is an isomorphism, and if gf = f', then HP(Z(f,X)) is isomorphic to HP(Z(f',X)). Proof: In the following analytic diagram cp' is the natural projection of X onto Z(f',X) and the function q from Z(f,X) to Z(f',X) is defined by q(z) = p'c-l(z). If f(x) = f(y), then f'(x) = gf(x) = gf(y) = f'(y) and so q is well defined. Since qcp = cp', q is continuous. q X > Z(f,X) -> Z(f',X) < X U k k' U f g I f A >Y > Y' A Define the function q' from (Z(f,X),k(Y)) to (Z(f',X), k'(Y)) by q'(x) = q(z). It follows from the Map Excision Theorem that q'* is an isomorphism. The function q" from k(Y) to k'(Y) is defined by q"(z) = q(z). Since q" = k'gk-1 and g* is an isomorphism, q"* is an isomorphism. Thus it follows from Lemma 1.1.2 that q* is an isomorphism. 1.3.5: Remark. If g is a continuous function from Y to Y', if h is a continuous function from Y' to Y, if gf = f', and if hf' = f, then Z(f,X) = Z(f',X). Proof: It is enough to show R(f,X) = R(f',X). If (x,y) e R(f,X), then x = y or f(x) = f(y). If f(x) = f(y), then f'(x) = gf(x) = gf(y) = f'(y). Similarly, R(f',X) is a subset of R(f,X). CHAPTER II HOMOTOPIC BORSUK'S PASTE JOBS HAVE THE SAME COHOMOLOGICAL STRUCTURE The main result of this chapter is that if X and Y are compact Hausdorff spaces, if A is a closed subspace of X, and if f and g are homotopic functions from A onto Y, then HP(Z(f,X)) is isomorphic to HP(Z(g,X)) for all non- negative integers p. By generalizing the definition of homotopic maps, it is possible to prove a more general theorem than indi- cated above. Section 0 contains the necessary definitions for this generalization. The proof of the theorem is di- vided into four parts, each of which constitutes a section in this chapter. Section 1 is devoted to a projection theorem; Section 2 contains a proof that if a Borsuk's Paste Job and a partial mapping cylinder have the same de- fining map, then they have isomorphic cohomology groups; Section 3 contains a technical proposition which shows that two particular partial mapping cylinders have iso- morphic cohomology groups; and Section 4 combines the pre- vious results to the main theorem. 0: T-Homotopy. Conventions: For the remainder of the chapter, r denotes a connected set. If H is a function from X x Y bo Z and y is a point in Y, then h denotes the function Erom X to Z defined by h (x) = H(x,y) for all x in X. The mnit interval is denoted by I. All spaces are Hausdorff. 2.0.1: Definition. Let f and g be functions from X to Y ind t and t' be points of T. Then f is T-homotopic to g it t, t' if and only if there exists a continuous function I from X x T to Y such that h = f and ht, = g. (If the pointss t and t' are not crucial, they will not be speci- Eied.) The space T will be called the connecting space. This idea arises from the fact that the Homotopy jemma [14] is nothing more than the Homotopy Theorem [14] stated for T-homotopic functions. The idea of T-homotopy las been used effectively in [15]. Some of the usual concepts involving I-homotopy can )e introduced into this context. ?.0.2: Definition. A subspace A of a space X is a T- leformation retract of X if and only if there exists a re- :raction of X onto A such that the retraction is T-homo- :opic to the identity. A space X is T-contractible to a point x from t on _ if and only if the injection of X onto X is T-homotopic it t, t' to the constant x-valued function on X. ?.0.3: Definition. Let P be a topological space. (a) An operation (multiplication) defined on P is a continuous function m from P x P to P. rhe pair (P,m) is a topploqical algebra. (b) Let m be an operation on P. Then a point p in P is a left unit for m if and only if m(p,x) = x for all x in P. A point p in P is a right unit for m if and only if m(x,p) = x for all x in P. A point p in P is a (two-sided) unit if and only if p is a left and a right unit for m. (c) Let m be an operation on P. Then a point p in P is a left zero for m if and only if m(p,x) = p for all x in P. A point p in P is a right zero for m if and only if m(x,p) = p for all x in P. A point p in P is a (two-sided) zero if and only if p is a left and a right zero for m. The connection between T-homotopy and topological al- gebra is quite strong. If T is a connected space and t,t' e T, then T is T-contractible to t from t' on t if and only if T admits an operation m such that t' is a right unit for m and t is a zero for m. These types of conditions are needed in this chapter and these restrictions are phrased in the language of topological algebra. Having made these definitions, the first question which arises is the question of how much of a generali- zation is T-homotopy over I-homotopy. The following ob- servations, which were made by Professor A. D. Wallace, will serve to illustrate that T-homotopy is a proper gen- eralization of I-homotopy. 2.0.4: Observation. If X is T-contractible, X is acyclic. 2.0.5: Observation. If T is a topological semi-lattice, then T is T-contractible. 2.0.6: Observation. If X is I-contractible to a point x, X is arcwise connected at x to every other point in X. 2.0.7: Observation. The Long Line L [15] is L-contract- ible but not I-contractible. Since the n-sphere is not acyclic, not every contin- uum is self-contractible. For the remainder of the chapter T is a connected space and t,t' c T. The next two lemmas demonstrate that for a "large" class of spaces T, the ideas of T-homotopy and I-homotopy coincide. A space P is arcwise connected between two points p and y of P if and only if there exists a homeomorphism h of I into P such that h(0) = p and h(l) = y. 2.0.8: Lemma. If T is arcwise connected between t and t', if f and g are mappings from X to Y, and if f is T-homo- topic to g at t,t', then f is I-homotopic to g. Proof: There exists a continuous function H from X x T to Y such that h = f and ht, = g. Since T is arcwise connected between t and t', there exists a function h from I to T such that h(0) = t and h(l) = t'. Let i be the in- jection of X onto X and define K from X x I to Y by K = H(i x h). Since K(x,0) = H(x,h(0)) = H(x,t) = f(x) and K(x,l) = H(x,h(l)) = h(x,t') = g(x) for all x in X, it follows that f is I-homotopic to g. 2.0.9: Lemma. If T is arcwise connected between t and t', if T is a normal space, and if f is I-homotopic to g, then f is T-homotopic to g at t,t'. Proof: There exists a continuous function h from I into T such that h(0) = t and h(l) = t'. The Tietze Extension Theorem states h may be extended to a function h' whose domain is T. Since f and g are I-homotopic, there exists a con- tinuous function H from X x I to Y such that h(x,0) = f(x) and H(x,l) = g(x) for all x in X. Define K from X x T to Y by K(x,t") = H(x,h'(t")). Clearly, K is continuous, K(x,t) = H(x,h(t)) = H(x,0) = f(x), and K(x,t') = g(x), so that f is T-homotopic to g. These two lemmas say that if T is arcwise connected and normal, then f is T-homotopic to g if and only if f is I-homotopic to g. Some properties of T-deformation retracts are studied in the remainder of this section. 2.0.10: Lemma. Let 0, 1 e T. If T is self-contractible to 0 from 1 on 0, then T x (0) is a T-deformation retract of T x T. Also, (0) x T is a T-deformation retract of T x T. Proof: The hypothesis ensures the existence of a function H from T x T to T such that hl(t) = t and h0(t) = 0 for all t in t. Define K from (T X T) x T to T x T by KU(t, t2),t3) = (tl,H(t2,t3)). The function K is clearly continuous. Thus kl(t1't2) = (tl,hl(t2)) = (tl,t2) so that kI is the identity on T x T. Since k0(tlt2) = (t,h0(t2)) = (t1,0), k0 is a retraction of T x T onto T x (0). It is well known that if X is a compact Hausdorff space and if A is a deformation retract of X, then the co- homology groups of X are isomorphic to those of A [14]. The same proof with minor changes will suffice to demon- strate the same result is true for T-deformation retracts. 2.0.11: Lemma. If A is a T-deformation retract of a com- pact Hausdorff space X and if i is the injection of A into X, then i* is an isomorphism. Proof: First it is shown that i* is a monomorphism. If g HP(X) such that i*(g) = 0, then there exists an open set M containing A such that j*(g) = 0 [14, Reduction Theorem] where M* denotes the closure of M and j is the injection of M* into X. If r is the injection of X into r* j* (X,M*), then HP(X,M*) --- HP(X)-- HP(M*) is exact so that there exists g' e HP(X,M*) such that r*(g') = g. Since A is a T-deformation retract of X, there exists a continuous function H from X x T to X and t,t' e T with the property that ht, is the identity function on X and ht is a retraction of X onto A. Define the continuous function f from (X,M*) into (X,X) by f(x) = ht(x). Since HP(X,X) = 0 [14], it follows that f* is identically zero. If q is the injection of X into (X,X), then fq(x) = ht(x) = rht(x) which implies that fq = rht and q*f* = (fq)* = (rht)* = ht*r*. Thus g = ht,*(g) = ht*(g) = ht*r*(g) = q*f*(g') = 0 and i* is a monomorphism. Since A is a re- tract of X, i* is an epimorphism [14, Proposition 53]. 2.0.12: Lemma. If A is a T-deformation retract of X, then HP(XA) is trivial for all non-negative integers po Proof: Let j be the injection of X into (X,A) and i be the injection of A into X. Since i* is an isomorphism and j* i* 0 -> H (X,A) -> H (X) -- H (A) is exact, it follows that 0 = K[i*] = I[j*]. Thus, the fact that j* is a monomor- phism implies H (X,A) = 0. i* 1 j* i* Since HP(X) -> HP(A) -> H (X,A) -- HP (X) -> HP+1{A) is exact and i* is an epimorphism, 6 is a zero map and j* is a monomorphism. But i* is a monomorphism so that j* is a zero map and hence HP(X,A) = 0. 2.0.13: Lemma. If A is a T-deformation retract of X, then there exists a continuous function f' from X to X such that f' is T-homotopic to the identity mapping on X, f'(X) = A and f' is idempotent. If the continuous function f from X to A is defined by f(x) = f'(x), then the induced homo- morphism f* is an isomorphism. Proof: If i is the injection of A into X, then by Lemma 2.0.11 the induced homomorphism i* from HP(X) to HP(A) is an isomorphism. Since if = f' implies f*i* = f'*, which is the iden- tity on HP(X), it follows that f* is an isomorphism. 1. A Projection Theorem. Notation: In the remainder of this chapter X and Y are compact Hausdorff spaces, A is a closed subspace of X, T is a continuum, and Z = (X x ftJ) U (A x T). In this section H is a continuous function from A x T onto Y. 2.1.1: Lemma. If ht is onto Y, then Z(H,St) is homeo- morphic to Z(ht,X). Proof: Let (p and cp' be the natural projections from 2t to Z(H,St) and from X to Z(ht,X), respectively, and let 9 be the restriction of cp to X x ([t. Define the continuous function m from X x (t) to X by m(x,t) = x. It will be shown that the equation k(z) = c'me (z) defines a homeo- morphism between Z(H,2t) and Z(ht,X). To see that k is well defined, it should be noted -1 that e (z) is not empty for all z e Z(H,Zt) and if 8(x,t) = 9(y,t), then x = y or ht(x) = H(x,t) = H(y,t) = ht(y) so that cp'm(x,t) = cp'm(y,t). To prove that k is continuous, it is sufficient to show that k- [B] = m-p'-l [B], for if B is closed, then em -1[B] is closed. If z e k-1B], then c'm- l(z) = k(z) e B so that z e em- -l[B]. Conversely, if z e em-1 C- [B], then k(z) = p'm-1 (z) e B and the equality follows. In order to show that k is a one-to-one function, let z,z' c Z(H,Zt) such that k(z) = k(z'). Then there exist points x,y e X with the property that R(x,t) = z and 9(y,t) = z'. If x # y, then x,y e A and co'(a) = k(z) = k(z') = c'(a) which implies that H(x,t) = ht(x) = ht(y) = H(y,t), and therefore z = 9(x,t) = 8(y,t) = z'. It remains only to show that k is onto. If z e Z(ht,X), then there is x e X such that z = c'(x). Since kcp(x,t) = z, k is onto. 2: The Cohomological Structures of Partial T-Mappinq Cylinders and Borsuk's Paste Job are the Same. Notation: Let 0 and 1 be two distinct points of T and let f be a continuous function from A onto Y. 2.2.1: Definition. Define the continuous function f' from A x (1) onto Y by f'(x,l) = f(x). The partial mapping T-cylinder of f over X at 0,1 is the Borsuk's Paste Job Z(f',o ). Since this partial mapping T-cylinder is deter- mined by f,X,0,1, and T, it is denoted by M(f,X,0,1,T) (or by M if it is clear what f,X,0,1, and T are). A partial mapping T-cylinder over X at 0,1 is called a partial mapping cylinder. This is the motivation for the definition which is given here. The reader is referred to [3] and [6] for more facts concerning partial mapping cy- linders. For the remainder of this chapter a standing hypo- thesis will be that there exists a continuous function H from T x T into T such that 0 is a zero for H and 1 is a unit for H. This assumption implies that T is T-contract- ible from 1 to 0. 2.2.2: Lemma. If p is a non-negative integer, then H(X x T, Zo) = 0. Proof: Define the continuous function K from (X X T x T, 20 x T) into (X X T, 2o) by K(x,tl,t2) = (x,H(t,t2)). Since kl(x,t) = K(x,t,l) = (x,H(t,l)) = (x,t), k1 is the identity map on (X x T, o ). But ko(x,t) = K(x,t,0) = (x,H(t,0)) = (x,0) so that k (X x T) is a subset of 2 . Thus k is a zero function [14, Corollary 1]. Since k O o is T-homotopic to the identity map on (X, Z0), k is the identity map on HP(X, Zo) and so H(X, 7 ) = 0. 2.2.3: Lemma. If m is the continuous function defined on X x T to X x (0) by m(x,t) = (x,0), then m* is an isomor- phism. Proof: First it will be shown that X x (0) is a T-defor- mation retract of X x T. The continuous function K which is defined by K(x,t,t') = (x,H(t,t')) is a function from X x T x T into X x T. Since kl(x,t) = (x,t) and k (x,t) = (x,0), it follows from Lemma 2.0.13 that m* is an isomor- phism. 2.2.4: Lemma. If v is a continuous function from 2 to 0 X x (0) defined by v(x,t) = (x,0) for all (x,t) e o, then v* is an isomorphism. Proof: If r is defined by r(x,t) = (x,t), then the follow- ing diagram is analytic: HP(X X T, 7o) HP(X x T) > HP o r* I v* H(X X (0), X x (0))- > HP(X x (0)) H(X x (0)) It follows that r* is an isomorphism since HP(X x T, Z ) = 0 = H(X X (0), X x (0)). If m is as in Lemma 2.2.5, then m* is an isomorphism. Thus, from Lemma 1.2.1, v* is also an isomorphism. 2.2.5: Lemma. If the function u from (Zo, A x (1) to (X x ({0, A x ({0) is defined by u(x,t) = (x,0), then u* is an isomorphism. Proof: If v is as in Lemma 2.2.4 and w is v restricted to A x (1), then the following diagram is analytic: H(X x (0) --> HP(A x (01) --> H (X x (01, A x (0)) v* J w* J u* H (o) > H (A x [11) > Hp+(, A x (1]) The mapping v* is an isomorphism by Lemma 2.2.4 and w* is also an isomorphism since w is a homeomorphism. It follows from Lemma 1.2.1 that u* is an isomorphism. No distinction will be made between X and X x ([0 since they are topologically the same. Similarly, no dis- tinction will be made between A and A x [01. 2.2.6: Proposition. The group HP(Z(f,X)) is isomorphic to the group HP(M(f,X,0,1,T)) for all non-negative integers p. Proof: Let Z = Z(f,X) and M = M(f,X,0,1,T). In the follow- ing analytic diagrams cp and c2 are the natural projections. X Z M CPo 2 P U k 1 k2 A > Y Ax [1} > Y The function h from M into Z is defined by h(z) = l 2-l (z),where v is as in Lemma 2.2.4. It is sufficient to show that h* is an isomorphism in order to establish the proposition. First it is seen that h is well defined and continuous. If c2(a,l) = cp(a',l) and a $ a', then f(a) = f(a') so that hc2(a,l) = coDv(a,l) = l(a,0) = co(a,0) = Cl(a',0) = cl1v(a',l) = hcD2(a',1) and hence h is well defined. Since he2 = c1, v is continuous, h is continuous. Because hk2(Y) is a subset of kl(Y), the function p which is defined by p(z) = h(z) is a mapping from (M,k2(Y)) to (Z,kl(Y)). If g denotes the restriction of h to k2(Y), then the following diagram is analytic: HP(M,k2(Y)) ---- HP(M) ---HP(k2(Y)) p* h* g* HP(Z,k,(Y)) > HP) -H(Z) HP(k(Y)) In order to establish that h* is an isomorphism, it is enough to show that p* and g* are isomorphisms. Since gco2(a,l) = lv2-12 c2(a,l) = cl(a) = k.f(a) = klk2-12(a,l), -i g = klk2 and so g* is an isomorphism since k* and k are isomorphisms. Define the function cl' from (X,A) to (X,k1 (Y)) by cp1 (x) = p(x) for all x C X and define the function 2 from (Zo, A X (1[) to (M,k2(Y)) by p2'(x) = p2(x) for all x e 2o. The Map Excision Theorem [13] says that c'* and c2'* are isomorphisms. If u is defined as in 2.2.4, then pz2'(x,t) = hcD2(x,t) = cplv(x,t) = pl'u(x,t) for all -l (x,t) e Z Thus, pc2 = cp'u, and so p* = (c2'*) -*(~')* which is an isomorphism. 3: A Relationship Between Two Partial Mapping T-Cylinders. The purpose of this section is purely technical. A relation between two partial mapping T-cylinders is needed to establish the main theorem and it is in this section that this relationship is obtained. For the remainder of this chapter it is assumed that there exists a continuous function K defined on T x T such that 1 is a zero for K and 0 is a unit for K. Notation: Let P be a continuous function from A X T onto Y, let M = M(P,2o,0,1), let M1 = M(P,2,0,1), let Z' = (X x (11 x ([0) U (A X T X T) = (Z1 x (0)) U (A x T x T), and let 0 = (X x (0) x (0)) U (A X T X T) = Z x (0) U (A x T x T). Let cp and cpl be the natural projections from o to M and from 1' to M1, respectively. The mappings p. (0,1) from (. ', A x T X T) to (Mi, Ai.) where A = ii(A X T x T) are defined by epi(x) = pi(x) where i = 0,1. It follows from the Map Excision Theorem [13] that cl'* and S'* are isomorphisms. The purpose of this section is to prove that HP(Mo) is isomorphic to HP(M ) for all non- negative integers p. The function p from 2 to 2 is defined by p(x,t,t') = (x,H(t,t'),t'). It is clear that p is continu- ous and that p(A x T x T) is contained in A x T x T. Hence, the continuous function j which is defined by j(x,t,t') = p(x,t,t') is a mapping from ( ', A x T X T) to (o, A x T X T). 2.3.1: Lemma. The mapping j* is an isomorphism from HP(Zo', A X T X T) onto HP(Z ', A x T x T). Proof: This lemma follows immediately from the Map Ex- cision Theorem [13]. Let q be the function from ML to Mo defined by q(z) -1 = c ppl (z) for all z c M1. It will be shown that q* is an isomorphism. 2.3,2: Lemma. The mapping q is well defined and continu- ous. Proof: If cpl(a,t,l) = co(a',t',l), then P(a,t) = P(a',t') so that by using the definition of p and the properties of H, it is seen that cpp(a,t,l) = p (a,H(t,l),l) = cp(a,t,l) = op(a',t',l) = cpo(a',H(t',l),l) = cop(a',t',l). Thus q is well defined. Since qcpl = cpop, qcp1 is continuous and it follows from Lemma 1.0.1 that q is also continuous. Since q(A1') is contained in A ', the continuous function r, defined by r(z) = q(z), is a mapping from (M1,A1') to (Mo',A ). 2.3.3: Lemma. The map r* is an isomorphism. Proof: From the definition of r, it is clear that rp, = Co'j so that cpI'*r* = (rcp1')* = (poj)* = *cpo'* Since *, j*, and C'* are isomorphisms, r* is an isomorphism. The function m from A to A is defined by m(z) = 1 o q(z). 2.3.4: Lemma. The set m(cpl(A x T X [1])) is contained in m(op (A X T X (11)). Proof: If (a,t,l) e (A x T x (11), then mcpI(a,t,l) = qcpl(a,t,l) = poP (a,t,l) = po(a,H(t,l),l) = cpo(a,t,l) e m(m,(A x T x (1)). For convenience, Ai" will denote m(e.i(A x T x 1i)) where i is either 0 or 1. From Lemma 2.3.4 it is clear that it is possible to define a continuous function m' from (A ', A,") to (A ', A") by m'(z) = m(z) for all z e A. The map m", which is the restriction of m to A ', is a continuous function from A to A I o 2.3.5: Lemma. The mapping m" is a homeomorphism. Proof: To see that m" is onto, let z e A ". Then there exists (a,t,l) e A x T x (1) such that a (a,t,l) = z. By using the appropriate definitions, it is seen that m"cpl(a,t,l) = qcpl(a,t,l) = cp (a,t,l) = o(a,H(t,l),l) = po(a,t,l) = z. It remains only to show that m" is a one-to-one function. If m"c~ (a,t,l) = m"cp (a',t',l), then by using the definitions of m, q, and p and by using the properties of H, it follows that po(a,t,l) = p o(a,H(t,l),l) = cop(a,t,l) = qco(a,t,l) = m"cp1(a,t,l) = m"cp1(a',t',l) = qcp (a',t',l) = pop(a',t',l) = cp~ (a',t',l). Therefore, P(a,t) = P(a',t') and thus pl(a,t,l) = cp (a',t',l). 2.3.6: Lemma. The map m'* is an isomorphism. Proof: Let p' be the continuous function defined from (A x T X T, A x T x (1i) to (A x T X T, A x T x {(1) by p'(a,t,t') = (a,H(t,t'),t'). Define the continuous func- tion L from (A x T x T x T, A x T x (11 x T) to (A x T X T, A x T X [11) by L(a,t,t',t") = (a,H(t,K(t',t")), t'). Since L(a,t,l,t') = (a,H(t,K(l,t')),l), the domain and range are correct. Since L(a,t,t',l) = (a,H(t,K(t',l)), t') = (a,H(t,l),t') = (a,t,t') and L(a,t,t',0) = (a,H(t,K (t',0)),t') = (a,H(t,t'),t') = p'(a,t,t'), p' is T-homo- topic to the identity on (A x T x T, A x T x (1]). Thus p'* is an isomorphism. Let ip and po" be the continuous functions from (A x T x T, A x T X (1)) to (Ao', A ") and (A ',A ), respectively, which are defined by i."(x) = oi(x), for i = 0,1. It follows immediately from the Map Excision Theorem that cp"* and c "* are isomorphisms. The following diagram is analytic : 4 A HP(Ao' ,Ao") H(A X T x T, A x T x I) om'* ITp Hp(A1',A1 ) -- HP(A x T x T, A x T (1) It is necessary to show that m' q" = p "p'. Clearly, m'ol" and ro "p' have the same range and domain. Since m'cl"(a,t,t') = cop(a,t,t') = co"(a,H(t,t'),t') 1 "p' (a,t,t'), cp m'* = p'*o and hence m'* is an isomor- phism because p'*,o "*, and c L"* are isomorphisms. 2.3.7: Lemma. The mapping m* is an isomorphism. Proof: Consider the following diagram: HP(Al',Al") >- HP(A') > H (A ") m"* m* Mi"* HP(AO',A") AI H (Ao') > HP (A ) where the unnamed mappings are the usual injections. The mappings m"* and m'* are isomorphisms so that it follows from Lemma 1.2.1 that m* is also an isomorphism. 2.3.8: Lemma. The mapping q* is an isomorphism. Proof: In the following diagram the unnamed mappings are injections. H (Mo,Ao) 0 HP(Mo) > H(Ao') jq* m* HP(M1,A ') > H (ML) > HP(Al) From Lemma 2.3.3, r* is an isomorphism and from Lemma 2.3.7, m* is an isomorphism. It follows that q* is also an isomorphism. To summarize the results of this section, the fol- lowing proposition is stated. 2.3.9: Proposition. Let T be a continuum and let 0,1 e T such that T admits a multiplication for which 0 is a zero and 1 is a unit and such that T admits a multiplication for which 1 is a zero and 0 is a unit. Let X and Y be compact Hausdorff spaces and let A be a closed subset of X. If P is a continuous function from A x T onto Y, then HP(M(O, P,0,1,T)) is isomorphic to HP(M(FI,P,O,1,T)). 4: The Main Theorem. In essence, the main theorem has already been proved; it remains only to state it and tie the loose ends together. 2.4.1: Proposition. Let X and Y be compact Hausdorff spaces, let A be a closed subset of X, and let T be a con- tinuum. If f and g are continuous functions from A onto Y which are T-homotopic at 0,1, and if T admits two opera- tions H and K such that 0 is a zero for H and a unit for K and such that 1 is a zero for K and a unit for H, then HP(Z(f,X)) is isomorphic to HP(Z(g,X)) for all non-negative integers p. Proof: Since f is T-homotopic to g at 0,1, there exists a continuous function P from A x T onto Y such that P(x,0) = f(x) and P(x,l) = g(x) for all x in A. From Lemma 2.1.1, it follows that H(Z(P,.o )) is isomorphic to HP(Z(f,X)) and IIP(Z(P,Zy)) is isomorphic to HP(Z{g,X)). From Proposition 2.2.5, HP(Z(P,I )) is iso- morphic to HP(M(P,Z1,0,1,T)) and HP(Z(P, o)) is isomorphic to HP(M{P,Z ,0,,T)). Therefore, by Proposition 2.3.9, HP(M(P,o,0, 1,T)) is isomorphic to HP(M(P,21,0,T)) and hence HP(Z(f,X)) is isomorphic to HP(Z(g,X)). The hypotheses of Proposition 2.4.1 are clearly sat- isfied by any topological lattice; in particular, these hy- potheses are satisfied by the unit interval. Thus, it fol- lows from Lemma 2.0.8 that if T = S then the conclusion of Proposition 2.4.1 holds even though the hypotheses of Proposition 2.4.1 are not satisfied by To avoid this Proposition 2.4.1 are not satisfied by S To avoid this type of example, Proposition 2.4.1 is re-stated in a slight- ly different form. 2.4.2: Theorem. Let X, Y, and A be as in Proposition 2.4.1 and T' be a continuum. If f and g are continuous functions from A onto Y such that f is T'-homotopic to g at 0, 1, and if there exists a subcontinuum T of T; with 0, 1 e T, and if T satisfies the hypotheses of Proposition 2.4.1, then HP(Z(f,X)) is isomorphic to HP(Z(g,X)) for all non-negative integers p. Proof: There exists a continuous function P from A x T' onto Y such that P(x,0) = f(x) and P(x,l) = g(x) for all x in A. Since restriction of P to A X T demonstrates the fact that f is T-homotopic to g at 0, 1, the conclusion of the theorem follows immediately from Proposition 2.4.1. CHAPTER III BORSUK'S PASTE JOB IN SEMIGROUPS If S is a semigroup, if A is a closed subsemigroup of S, and if f is a continuous epimorphism defined on A, then the question arises as to whether Z(f,S) admits a natural semigroup structure. To be more explicit, is R(f,S) a congruence of S? The answer is sometimes. Thus, the problem which is studied in this chapter is to find conditions on S and A which will insure that R(f,S) is a congruence of S for all epimorphisms f defined on A. Two propositions of this type will be proved in Section 2. In Section 1 there is a more technical discussion of the prob- lem and an indication of why certain hypotheses are neces- sary. A simple example which is contained in Section 3 shows that there exists a semigroup X and an epimorphism g defined on the minimal ideal of X such that R(g,X) is not a congruence of X. Since this example consists of left ze- ros, the case in which the subsemigroup A consists of left zeros is investigated in Section 4. The main result of these investigations is a theorem which gives a set of necessary and sufficient conditions for R(f,S) to be a congruence of a semigroup S for all epimorphisms f defined on a subsemigroup A of S in the case that A consists of left zeros. In Section 5 this theorem is extended to ob- tain a set of necessary and sufficient conditions for R(f,S) to be a congruence for all epimorphisms f defined on the minimal ideal. In Section 6 there is presented a more general version of the theorems of Sections 4 and 5. The theorem in Section 6 gives necessary and sufficient conditions for R(f,S) to be a congruence of S for all epimorphisms defined on a subsemigroup A. This characterization is in terms of the congruences of A. 0: Notation. A semiqroup is a non-empty Hausdorff space together with a continuous, associative multiplication. Precisely, a semigroup is a non-empty Hausdorff space S and a function m from S x S to S that satisfies the following conditions: (1) S is a non-empty Hausdorff space, (2) m is continuous, (3) m is associative, i.e., m(x,m(y,z)) = m(m(x,y),z) for all x,y,z e S. For convenience, m(x,y) will be denoted by xy and following customary usage, we shall say "S is a semigroup" when it is clear what the multiplication for S is. In this chapter S is a semigroup. If A and B are subsets of S, then AB = (abla c A and b e B). A non-empty subset L of S is a left ideal of S if and only if SL is a subset of L. A non-empty subset R of S is a right ideal of S if and only if RS is a subset of R. A non-empty subset I of S is a (two-sided) ideal of S if and only if SI U IS is a subset of I. A compact semigroup has minimal left, right, and two-sided ideals. The mini- mal ideal, if it exists, is unique and is denoted by K. A point e of S is said to be idempotent if and only 2 if e = ee = e. Let E be the set of idempotents of S. It is known that if e e E, then there exists a maximal sub- group H of S which contains e. If e E E n K, then eSe = He, eS is a minimal right ideal, Se is a minimal left ideal, and eS n Se = eSe. Also, K = U(eSele e E n K) = U(RIR is a minimal right ideal of S} = U(LIL is a minimal left ideal of S). See [2], [10], and [15]. 1: Preliminary Propositions. 3.1.1: Definition. If S is a semigroup and if F is a closed equivalence relation defined on S, then F is a congruence of S if and only if (AS)F U F(AS) is a subset of F, where AS is the diagonal of S x S. The following lemma is well known [15]. 3.1.2: Lemma. Let S be a compact semigroup and let F be a closed equivalence relation defined on S. Then F is a congruence of S if and only if S/F admits a unique continu- ous, associative multiplication m* such that m*(cp x p) = mc where p is the natural projection of S onto S/F and m is the multiplication defined on S. S/F X S/F S/F Ixcp I S x S S The set S/F is said to admit a desirable multipli- cation if and only if F is a congruence of S. Let S and T be compact semigroups, let A be a closed subset of S, and let f be a continuous function defined on A. The problem studied in this chapter is to determine when R(f,S) is a congruence of S. The following lemma will give a criterion which is useful in studying this problem. 3.1.3: Lemma. Let S and T be compact semigroups, let A be a closed subset of S, and let f be a continuous function from A onto T. Then R(f,S) is a congruence of S if and only if (3.1) is satisfied. (3.1) If x e S, if a,a' e A, and if f(a) = f(a'), then f(xa) = f(xa') or xa = xa', and f(ax) = f(a'x) or ax = a 'x. Proof: The necessity is obvious. Sufficiency: If x e S and (y,z) e R(f,S), then f(y) = f(z) or y = z. If y = z, then xy = xz and yx = zy so that (xy, xz) and (yx, zy) are in R(f,S). If f(x) = f(z), then it follows from (3.1) that (yx, zx) and (xy, xz) are in R(f,S). 3.1.4: Remark. If S, A, T, and f are as in Lemma 3.1.3, and if f is a one-to-one function, then R(f,S) is a congru- ence of S. 3.1.5: Lemma. If S, A, T, and f are as in Lemma 3.1.3, if p is the natural projection, if R(f,S) is a congruence, and if k is the unique homeomorphism such that kf is the re- striction of co to A, then kf(xy) = kf(x)kf(y) whenever x,y,xy e A. S Z(f,S) U k f I A > T 3.1.6: Remark. If S, A, T, f, cD, k, and R(f,S) are as in Lemma 3.1.5, then k is an is iseomorphism (both an isomor- phism and a homeomorphism) if and only if f is a homomor- phism and A is a subsemigroup of S. Proof: Assume A is a subsemigroup of S and suppose that f is a homomorphism. If y, y' e T, then there exist x, x' e A such that y = f(x) and y' = f(x'). It follows that k(y)k(y') = k(f(x))k(f(x')) = kf(xx') = k(f(x)f(x")) = k(yy') and thus k is homomorphism. Since k is a homeomor- phism, k is an iseomorphism. If k is an iseomorphism, then k(T) = cr(A) is a sub- -l semigroup and so A = m- o(A) is a subsemigroup of S. Also, if x,y e A, then f(x)f(y) = k- (x)k l(y) =k ((x)-o(y)) = k- o(xy) = f(xy) since k-1 and c are homomorphisms. Let S and T be compact semigroups, let A be a closed subset of S, and let f be a continuous function from A onto T. If R(f,S) is a congruence of S, then it is desirable to be able to embed T iseomorphically into Z(f,S) by a function k such that k- = f, where -o is the natural projection of k such that k z~ = f, where r is the natural projection of S onto Z(f,S) restricted to A. In view of the above re- marks and Remark 3.1.6, we shall restrict our attention to the case when A is a subsemigroup of S and f is a homomor- phism. It will be shown that it is also necessary to re- strict our attention more by requiring A to be similar to an ideal. This is the reason for the next definition. Let S be a semigroup and let B be a subset of S. Then B(-1) B = (x c SIBx n B rn), B B(-1) = x c SjxB n B Sr3), BI-1] B = (x e SIBx c B), and B B[-1] = (x c SxB c B). 3.1.7: Definition. Let S be a semigroup and let A be a non-empty subset of S, then A is a semi-ideal of S if and only if (i) A is closed, (ii) If x e S, then card[xA n (S--A)] e 1 and card[Ax n (S--A)] A 1, and (iii) A ()A c A[-]A and AA(-1) A[-1] AA- The following remark gives a few of the properties of semi- ideals. 3.1.8: Remark. (a) Let A be a closed subset of a semi- group S, then A is a semi-ideal of S if and only if (i) If x e S, then card xA = 1 or x e AA and (ii) If x e S, then card A = 1 or x e A[-1]A. (b) A closed ideal of S is a semi-ideal of S. (c) If A is a semi-ideal of S and if A contains an ideal of S, then A is an ideal of S. (d) If G is a topological group and if A is a closed subset of G, then A is a semi-ideal of G if and only if card A = 1 or A = G. (e) The intersection of a filter base of semi-ideals of a compact semigroup is a semi-ideal of that semigroup. (f) If B is a subset of a compact semi- group, then there exists a semi-ideal minimal with respect to containing B. (g) Semi-ideals are preserved under closed epimorphisms. (h) Semi-ideals are not necessarily pre- served under inverses of continuous epimorphisms. Proof: (a) If A is a semi-ideal of S and if x e S, then x (S-A(-)) r AA-]. If x e (S--AA(-)), then xA c(S--A) so that card xA = card[xA n(S--A)] = 1. The proof of (ii) follows in a similar fashion to that of the proof of (i). Suppose (i) and (ii) hold and let x e AA(-l). It follows that x C AA[-l] or card xA = 1. If card xA = 1, then x e AA1-l] so that AA(-l) is a subset of AA[-]. It is clear that the other parts of the definition of a semi- ideal are also satisfied. (b) is clear. (c) If I c A, if A is a semi-ideal of S, if I is an ideal of S, and if x c S, then ] I n Ix c A n AX so that x e AA[-1] and A is a right ideal. Similarly, A is a left ideal. (d) The sufficiency is clear. Necessity: If card A > 1, then there exists a point x in A such that x is distinct from the identity e of G. Since card x A is greater than 1, x A is a subset of A and so e is an element of A. If y e G, then the cardinal- ity of yA is greater than one. Hence y e yA c A and there- fore A = G. (e) Let T be a filter base of semi-ideals of a compact semigroup S, then B = n(B'IB' e ]3 is closed. If x e BB(-), then r] xB n B c xB' n B' for all B' e 3. Since each B' is a semi-ideal, x e B'B'[-1] so that xB c xB' c B'. Thus, xB c n(B'IB' e 3! = B so that BB(-1) c BB[-l] If x e S BB(-), then there exists B e 3 such that xB n Bo = For suppose xB' n B' o Q for all B' c 8, then C (xB' n B'IB' e ~) = (nfxB'IB' E B)) n (n[B'IB e 3)) c xB n B so that x e BB(-1) which is a con- tradiction. Right multiplication follows in a similar fashion. A straight-forward application of the Hausdorff Maximality Principle will prove (f). (g) Let S be a semigroup, let f be a closed eipmorphism from S onto a semigroup T, and let A be a semi-ideal of S. If f(x) c (f(A))(f(A) (1)), then xA c A or card xA = 1. If xA c A, then f(x)f(A) = f(xA) c f(A) so that f(x) c f(A)(f(A) -). If card xA = 1, then card f(x)f(A) = 1 and f(x) c f(A)(f(A)[-1]). Suppose f(x) c (T [(f(A))(f(A) [ )]), then f(xA n A) c f(xA) n f(A) = r so that x C (S AA'- ). This means that card xA = 1 and therefore card f(x)f(A) = 1. Right multiplication is verified in a similar fashion. (h) Let S and T be the semigroup of com- plex numbers whose modulus is one under the usual multi- 2 plication. Define f from S to T by f(z) = z. It follows from (d) that each point w of T is a semi-ideal, but f- (w) is not a semi-ideal. The following lemma is a re-statement of Lemma 3.1.3. 3.1.9: Lemma. Let S and T be compact semigroups, let A be a subsemigroup and semi-ideal of S, and let f be a continu- ous epimorphism from A onto T. Then R(f,S) is a congruence of S if and only if (3.1.9.1) x c [(AA 1 ) A] and f(a) = f(a') imply f(xa') = f(xa) and (3.L.9.2) x e [(A [- A)] and f(a) = f(a') imply f(ax) = f(a'x). Proof: Sufficiency. It is enough to verify 3.1. If x c S, then x E A or x e [S (AA- )] or x e [(AA- A]. Suppose a,a' c A and f(a) = f(a'), then if x A it fol- lows that f(xa) = f(x)f(a) = f(x)f(a') = f(xa'). If x r (S (A[- A)), then xa = xa' since card xA = 1. If x e [(AA I]) A], it follows from 3.1.9.1 that f(xa) = f(xa'). The other part of 3.6 follows in similar fashion. The necessity is obvious. This lemma makes explicit the following idea. If A is a subsemigroup and a semi-ideal of S, then S is parti- tioned into three sets: A, [(A [ A) A] and [S - (A (-A)]. It is easy to see that if x is in A U (S A(-1A), then (x,x)-Z(f,S) is a subset of Z(f,S). In order to say that Z(f,S) is a congruence, one must, how- ever, know what the behavior is of the points in [(A[- ]A) - A]. Similar statements can be made about right multi- plication. Semi-ideals arose out of the necessity of having a subset satisfy conditions similar to the conditions for being an ideal. It will be shown that the property of be- ing a semi-ideal is the "weakest" assumption that will work for the theorems presented here. 3.1.10: Definition. Let A be a closed subsemigroup of a compact semigroup S, let T be a degenerate semigroup, and let f be a continuous epimorphism from A onto T. The space Z(f,S) is called the Generalized Rees Quotient of S by A and is denoted by Q(A,S). Equivalently, Q(A,S) = S/R where R = (A x A) U (AS). The Generalized Rees Quotient is an extreme case of Borsuk's Paste Job and as such is helpful in determining restrictions which should be placed on A. 3.1.11: Proposition. Let A be a closed subsemigroup of a compact semigroup S. Then Q(A,S) admits a desirable multiplication (i.e., (A x A) U (AS) is a congruence of S) if and only if A is a semi-ideal of S. Proof: Let T and f be as in Definition 3.1.10. Necessity. Suppose x e AA(-), then there exists a point y in A such that xy e xA n A, then xy xz. This implies that (3.1) is not satisfied, which is a contradic- tion. Thus AA is a subset of AA[-]. Suppose x e S and y,x c A such that xy,xz c (S-A). According to (3.1), xy = xz so that the cardinality of xA n (S-A) is no greater than 1. Right multiplication may be verified in a similar fashion. Sufficiency. It is enough to verify 3.1.9.1 and 3.1.9.2. If x r [(AA[-1 ) A] and a,a' e A, then f(xa) f(xa') since the range of f is degenerate. The property 3.1.9.2 may be verified in a similar fashion. 3.1.12: Definition. If A is a compact subsemigroup of a compact semigroup S, then A is agreeable with respect to S if and only if R(f,S) is a congruence of S for every con- tinuous epimorphism f defined on A. Thus it is now possible to restate the problem which is studied in this chapter in technical language. The problem is to find conditions which will insure that a sub- semigroup is agreeable. 3.1.13: Remark. (a) If S is a compact semigroup and if a is an idempotent, then (a) is agreeable with respect to S. (b) If S is a compact semigroup, then S is agreeable with respect to S. 3.1.14: Remark. If A is a closed subsemigroup of a com- pact semigroup S, then A is agreeable with respect to S if and only if F U (AS) is a congruence of S for every con- gruence F of S. Proof: Necessity. Suppose F is a closed congruence of A. If f is the natural projection of A onto A/F, then R(f,S) = (F U AS) is a congruence of S since A is agreeable with respect to S. Sufficiency. If f is a continuous epimorphism from A onto T, then [(f x f)-1(AT)] U AS = R(f,S), and since [(f x f)- (AT)] is a congruence of S, then R(f,S) is a congruence of S. 3.1.15: Proposition. Let A be a closed subsemigroup of a compact semigroup S. If A is agreeable with respect to S, then every semi-ideal of A is a semi-ideal of S. Proof: If P is a semi-ideal of A, then the natural pro- jection f of A onto Q(P,S) is a continuous epimorphism. Since A is agreeable with respect to S, Z(f,S) = Q(P,S) admits a desirable multiplication and it follows from Pro- position 3.1.11 that P is a semi-ideal of S. It will be shown later that the converse to this proposition is not true. However, the converse is true in a certain special case which will be investigated in Section 4. 2: Sufficient Conditions. 3.2.1: Definition. Let S be a semigroup and let r be a function from S to S. Then r is a homomorphic retraction if and only if r is a homomorphism and a retraction, and r(S) is a homomorphic retract of S whenever r is a homo- morphic retraction of S. In the remainder of this section S is a semigroup and A is a subsemigroup and semi-ideal of S. 3.2.2: Lemma. The sets AA[-l] and A[-1]A are subsemi- groups of S which contain A as a left ideal and as a right ideal, respectively. Proof: If x,y e AA-l], then xyA = x(yA) c xA c A so that AA[-] is a subsemigroup of S and A is a left ideal of AA[-l] The proof for A [1]A is similar. 3.2.3: Proposition. If A is a homomorphic retract of AA-] and A A, then A is agreeable with respect to S. Proof: It is enough to show that 3.1.9 is satisfied for an arbitrary continuous epimorphism f defined on A. Let r be a homomorphic retraction from A[- A onto A. If x e [(A[-1 A) --A] and a,a' c A such that f(a) = f f(a'), then, using the properties of a homomorphic retrac- tion and the fact that A is an ideal in A[- ]A, it is seen that f(ax) = f(r(ax)) = f(r(a)r(x)) = f(ar(x)) = f(a)f(r(x)) = f(a')f(r(x)) = f(a'r(x)) = f(r(a')r(x)) = f(r(a'x)) - f(a'x). The property 3.1.9.1 follows in a similar fashion. Anticipating the results of Chapter IV, it may be seen that conditions on A, such as A contains a unit, will insure that A is a homomorphic retract of A[-1]A and AA Using this result, the following corollary is obtained. 3.2.4: Corollary. If the minimal ideal of S is a group, then the minimal ideal is agreeable with respect to S. The following proposition will be useful in Sections 4 and 5. 3.2.5: Proposition. If A satisfies the following condi- tions, then A is agreeable with respect to S. (1) If x e [(AA[11]) A], then card xA = 1 or xa = a for all a in A. (2) If x e [(A[-1]A) -A], then card Ax = 1 or ax = a for all a in A. Proof: It is enough to verify 3.1.9 for an arbitrary epi- morphism f defined on A. If x e[(A -A) A] and if a,a' c A such that f(a) =f(a'), then card Ax = 1, or ax = a, and a'x = a'. If card Ax = 1, then ax = a'x. If ax = a and a'x = a', then f(ax) = f(a'x). The verification of 3.1.9.1 is similar. 3: Examples. In this section, S will always be a semigroup. The following lemma will be useful in this section. 3.3.1: Lemma. (a) If A is a closed subset of S and if S consists of left zeros, then A is a semi-ideal of S. (b) If A is a closed subset of S and S consists of right zeros, then A is semi-ideal of S. Proof: If x e S, then xA = x so that the cardinality of xA is one. Since Ax = A, A- A = S. The proof of (b) is similar to that of (a). 3.3.2: Example. This is an example of a semigroup with a semi-ideal which is not an ideal but which is agreeable. Let S be a compact Hausdorff space with the multi- plication defined by xy = x and let A be a proper subset of S. Since A is a proper subset of S, there exists a point z in S which is not in A so that zA = z, which means A is not an ideal. It follows from Lemma 3.3.1 that A is a semi-ideal and it follows from Proposition 3.2.5 that A is agreeable. 3.3.3: Example. This is an example of a semigroup whose minimal ideal satisfies the hypotheses of Proposition 3.2.5. Let J be the unit interval with the multiplication defined by xy = x, let P be f0,1) with the usual multipli- cation, and let S be J x P with the co-ordinatewise multi- plication. The minimal ideal of S is J x [0). It is easi- ly verified that the hypotheses of Proposition 3.2.5 are satisfied. The following example is due to P. E. Connor. 3.3.4: Example. Let X be the unit interval with the usual multiplication, let Y be the space of complex numbers with norm one endowed with the usual multiplication, let S be X x Y endowed with co-ordinatewise multiplication, and let f be the continuous function from (0 x Y to Y defined by 2 f(O,z) = z The relation R(f,S) is a congruence of S and Z(f,S) is topologically a M6bius Band. 3.3.5: Example. This is an example of a semigroup whose minimal ideal is not agreeable. Let S be the triangle semigroup, i.e., S = {x Y]Ix and y are real non-negative numbers such that x + y ; 1. The minimal ideal K is [0 fll y is a real number satisfy- ing 0 a y S 1}. Since K consists of left zeros, any sub- set of K is a semi-ideal of K; in particular, let A = {Fo0 1? Since F o OiF i1 = 0 '7 and [1 0][0 l [0 '] then A is not a semi-ideal of S. Thus it follows 0o 1J, from Proposition 3.1.15 that K is not agreeable with re- spect to S. 3.3.6: Example. This is an example of a semigroup T which is topologically a unit interval and such that the minimal ideal of T is not agreeable with respect to T. Let S, K, and A be as in Example 3.3.5 and let T = K U {[x O x is a real number satisfying 0 d x < 1l. The minimal ideal of T is K and A is a semi-ideal of K but not of T. It follows that K is not agreeable with respect to T. 3.3.7: Example. This example will serve to demonstrate that the converse of Proposition 3.1.15 is, in general, false. Let T be as in Example 3.3.6, let Y = [0,1] with the multiplication defined by xy = y, and let S = T x Y with co-ordinatewise multiplication. The minimal ideal K' of S is K x Y where K is the minimal ideal of T. (3.3.7.1) If A is a semi-ideal of K, then card A = 1 or A = K. Proof: Let A be a semi-ideal of K and suppose card A > 1. It follows that there are two distinct points, (x,y) and (u,w), in A and hence two cases to be considered: (1) x 4 u and (2) y # w. (1) x 4 u. If z e Y, then, using the fact that K' consists of left zeros, it is seen that (x,y)(x,z) = (x,z) 4 (u,z) = (u,w)(x,z) = (u,x)(x,z). Thus, since card A(x,z) > 1 and since A is a semi-ideal of K, then (x,z) c A[ A so that (u,z),(x,z) e A. This means (fx x Y) U (u x Y) c K. If z c K', t C Y, and t' e Y such that t' 4 t, then (x,t')(x,t) e A. But (z,t)(x,t) = (z,t) (z,t) = (z,t) (x,t') so that card (z,t)A > 1. It follows from the fact that A is a semi-ideal of K that (z,t) e AA[-]. Thus, (z,t) e (z,t)A C A which means K c A. (2) y w. This argument is dual to that in case (1). It is now clear that every semi-ideal of I is a semi-ideal of S. It remains only to show that K is not agreeable with respect to S. Define the continuous function f' from K' onto K' by f 0< x [1 -= 2 if x < j and [01 if x 2 De- fine the continuous function f" from Y onto Y by f"(x) = zx if x s j and 1 if x > J. Define f from K to K by f(x,y) = (f'(x),f"(y)). The function f is clearly continuous and a simple computation shows it is a homomorphism. It remains yet to show that (3.1) is not satisfied. Let x = [0 y= [ z = l, and t = It follows that f(x,0) = (y,0) = f(y,O) t f(t,0) but (z,0) (y,0) = (z,0) and (z,0)(x,0) = (t,0) so that (3.1) is not satisfied. 4: The Case in Which A Consists of Left Zeros. Throughout this section, S is a semigroup and A is a closed subsemigroup of S. The agreeability of A with re- spect to S is investigated in the case where A consists of left zeros. It turns out that a condition weaker than A consisting of left zeros is needed for this investigation; this condition is called 2-simplicity. A theorem in this section gives necessary and sufficient conditions for a 2- simple closed subsemigroup to be agreeable. 3.4.1: Definition. A semigroup T is n-simple if and only if the cardinality of T is not less than n and every sub- set of T with n elements is a semi-ideal of T where n > 0. The semigroup is totally simple if and only if whenever n is not greater than the cardinality of T, then T is n- simple. A few remarks concerning the structure of 2-simple semigroups are needed before the proof of the main result of this section. The following lemma will demonstrate that the idea of total simplicity is a generalization of the ideal of semigroups consisting of left or right zeros. 3.4.2: Lemma. If T is a semigroup such that xT = x or Tx = x for all x e T, then T is totally simple. Proof: Suppose n card T and let B be a subset of T with cardinality n. If x e T, then xB = x(Bx = x) so that card [xB n (T B)] s 1 (card[Bx n (T B)] 9 1). Since Bx = B(xB = B), x e B[-1]B (x c BB[-1]) The type of homomorphisms which may be defined on a 2-simple semigroup is described in the next lemma. 3.4.3: Lemma. If P is a semigroup of cardinality of at least two, then P is 2-simple if and only if for every pair x,y of distinct points of P there exists a semigroup T and a continuous epimorphism f from P to T with the properties that f(x) = f(y), f restricted to P (x,y) is a mono- morphism, and f[P -- x,y]] = T -- f(x)l. Proof: Suppose P is 2-simple and x,y are two distinct points of P. Since (x,y) is a semi-ideal, it follows from Proposition 3.1.11 that Q(fx,y),P) admits a desirable mul- tiplication. If f is the natural projection from P onto Q((x,y),P), then f is easily seen to have the required pro- perties. Suppose x,y e P and let T and f be as described. Since P is agreeable with respect to P, Z(f,P) admits a desirable multiplication. But Z(f,P) = Q((x,y),P), which according to Proposition 3.1.11 admits a desirable multi- plication if and only if fx,y) is a semi-ideal of P. Thus, P is 2-simple. 3.4.4: Remark. A group is 2-simple if and only if it is the cyclic group of order 2. Theorem 3.4.9 gives a set of necessary and sufficient conditions for a 2-simple semigroup to be agreeable; this theorem is the main result of this section. The main part of the proof to Theorem 3.4.9 is presented in Lemmas 3.4.5, 3.4.6, and 3.4.7. 3.4.5: Lemma. Let A be a subsemigroup of a semigroup S. If A is a semi-ideal of S and A satisfies conditions (1) and (2), then every semi-ideal of A is a semi-ideal of S. (1) If x e AA-1, then card xA = 1 or xa = a for all as A. (2) If x e A[-1]A, then card Ax = 1 or ax = a for all a A. Proof: This follows from Proposition 3.2.5 and 3.1.16. 3.4.6: Lemma. Let A be a 2-simple subsemigroup of a semi- group S. (i) If there are x e AA[-1] and a e A such that xa $ a and card xA > 2, then there exists a semi- ideal of A which is not .a semi-ideal of S. (ii) If there is a point x in A[-1]A and a point a in A such that ax / a and card Ax > 2, then there exists a semi-ideal of A which is not a semi-ideal of S. Proof: (i) Since card xA > 2, there exists a' e xA such that xa $ a' and a' 4 a. From the fact that a' e xA, it follows that there is a" c A with the property xa" = a'. Let B = (a,a"); B is a semi-ideal of A because card B = 2. Suppose B is a semi-ideal of S. Using the fact that xa f a' = xa", it is seen that card xB = 2 so that x e BB[-1 This implies that xa, xa" e B and xa = a" since x f a. Also, it follows that xa" i a, for if xa" = a, then a' = a and a' was chosen so that a' a. Therefore, xa" = a" which implies a" = a' since a' = xa". Thus, xa = a" = a', but this is a contradiction since a' was chosen in such a manner that xa a', and therefore B is not a semi-ideal of S. The proof of (ii) follows in a similar fashion. 3.4.7: Lemma. Let A be a 2-simple subsemigroup of a semi- group S and let the cardinality of A be greater than two. (i) If there exists a point x in AA[-1] such that card xA = 2, then there is a semi-ideal of A which is not a semi-ideal of S. (ii) If there exists a point x in A[- ]A such that card Ax = 2, then there is a semi-ideal of A which is not a semi-ideal of S. Proof: Since xA is a proper subset of A, there is a point a in A xA; let xa = a'. There exists a point a' in xA (a'l since card xA = 2. Thus a,a',a" are all dis- tinct points and xA = fa',a"). There are three cases to be considered: (1) xa' = a", (2) xa' = a', xa" = a", and (3) xa' = a', xa" = a'. (1) xa' = a". If B = fa,a'), then since A is 2- simple, B is a semi-ideal of A. If B is also a semi-ideal of S, then since xa = a' e B, x e BB(-) c BB- Thus, a" = xa' C B but a' 4 B, which is a contradiction. There- fore B is not a semi-ideal of S. (2) xa' = a' and xa" = a". If B = (a,a"), then B is a semi-ideal since A is 2-simple. Since xa" = a", x e BB-1). But xa = a' e B so that x BB-I]. Thus BB(-1) is not a subset of BB -l] and B is not a semi- ideal of S. (3) xa' = a' and xa" = a'. Since a" is in xA, there is a point a"' in A such that xa"' = a". Since a' i a", a"' a. If B = (a,a"'), then the 2-simplicity of A implies B is a semi-ideal of A. Since xa = a' and xa"' = a", it fol- lows that xB=(S B). But card xB = 2 so that B is not a semi-ideal of S. The proof of (ii) is similar to the proof of (i). 3.4.8: Theorem. Let A be a 2-simple subsemigroup of a semigroup S such that the cardinality of A is larger than 2. Then every semi-ideal of A is a semi-ideal of S if and only if the conditions (1), (2), and (3) are satisfied. (1) If x e AA[-], then card xA = 1 or xa = a for all a in A. (2) If x e A -A, then card Ax = 1 or ax = a for all a in A. (3) A is a semi-ideal of S. Proof: This theorem follows from Lemmas 3.4.5, 3.4.6, and 3.4.7. 3.4.9: Theorem. Let A be a 2-simple compact subsemigroup of a compact semigroup S such that the cardinality of A is greater than two. Then A is agreeable with respect to S if and only if every semi-ideal of A is a semi-ideal of S. Proof: Necessity. This follows immediately from Proposi- tion 3.1.15. Sufficiency. Theorem 3.4.8 says that the hypotheses of Proposition 3.2.5 are satisfied. 3.4.10: Corollary. Let A be a compact 2-simple subsemi- group of a compact semigroup S. If the cardinality of A is greater than two, then A is agreeable with respect to S if and only if the conditions (1), (2), and (3) are sat- isfied. (1) If x e AA [], then card xA = 1 or xa = a for all a in A. (2) If x e A(-I]A, then card Ax = 1 or ax = a for all a in A. (3) A is a semi-ideal of S. 3.4.11: Corollary. A semigroup T with cardinality great- er than two is 2-simple if and only if the following two conditions are satisfied. (1) If x e TT[-l], then card xT = 1 or xa = a for all a in T. (2) If x e T[-1]T, then card Tx = 1 or ax = a for all a in T. Proof: The necessity is immediate from Theorem 3.4.8 with T = S = A. Sufficiency. Suppose a and a' are distinct points of T, let x e T and let B = (a,a'). If card xA = 1, then card xB = I. If card xA = 1, then xB = B so that x c BB[-1] The argument for right multiplication is similar. 3.4.12: Corollary. Let A be a closed subsemigroup of a compact semigroup S. If the cardinality of A is larger than two and if A consists of left zeros of S, then A is agreeable with respect to S if and only if x e AA[-1] im- plies card xA = 1 or xa = a for all a in A. Proof: Using Lemma 3.4.2, it is seen that A is 2-simple, and so Corollary 3.4.10 is applicable. Since A consists of left zeros of S, (2) of 3.4.10 is satisfied. For right zeros, a statement similar to 3.4.12 is made. 3.4.13: Corollary. Let A be a compact subsemigroup of a compact semigroup S. If the cardinality of A is greater than two and if A consists of right zeros of S, then A is agreeable with respect to S if and only if x e A -1]A im- plies card xA = 1 or ax = a for all a in A. 5: Agreeable Minimal Ideals. In this section, S is a compact semigroup and K is the minimal ideal of S. A theorem which gives necessary and sufficient conditions for K to be agreeable with respect to S will be proved in this section. Various structure theorems for K will be used in this section. These theo- rems may be found in [2], [10], [16], [18], and [20]. 3.5.1: Lemma. Let A be a compact subsemigroup of S, let f be a continuous epimorphism from A onto P, let cp be the natural projection of S onto Z(f,S), and let k be the natural embedding of P into Z(f,S). If A is agreeable with respect to S, then k(P) is agreeable with respect to Z(f,S). Moreover, if g is a continuous epimorphism from P -l onto T, then g' = gk and if p' and c" are the natural projections of Z(f,S) and S onto Z(g', Z(f,S)) and Z(gf,S), respectively, then there exists an iseomorphism h from Z(gf,S) into Z(g',Z(f,S)) S > Z(f,S) S > Z(gf,S) A > P A > T S > Z(f,S) Z(g',Z(f,S)) U U k' kf g' A t k(P) T Proof: Since semi-ideals and semigroups are preserved under epimorphisms, k(P) = c(P) is a semigroup and semi- ideal. Thus it is enough to verify (3.1) for a continuous epimorphism g' from k(P) onto T. Suppose c(x) c [k(P)1 [k(P)] and g'co(a) = g'cp(a'). Since A is agreeable with re- spect to S, (3.1) must hold for the epimorphism g'p. Thus g'[p(x)c~(a)] = g'cp(xa) = g'cp(xa') = g'[cp(x)cp(a)] and k(P) is agreeable with respect to Z(f,S). Let h be defined by h(z) = cp'M" -(z). It has been shown previously that h is a homeomorphism, so that it is enough to verify that h is a homomorphism. Using the homo- morphic properties of p, cp', and p", it is seen that h[cp"(x) p"(y) ] = hp"(xy) = cp'(xy) = p'cp(x)cp'p(y) = h(p"(x))h(cp"(y)). Notation: The function u from K into E is defined by u(x) = e if and only if x c eS n Se. Wallace [20] has shown that u is continuous. Through Proposition 3.5.4 e will be a fixed point in K n E, P will denote Se n E, and f will be the function from K to P defined by f(x) = u(xe). It should be noted that P is a subsemigroup which is 2-simple, since pq = p for all p,q e P. 3.5.2: Lemma. The function f is a homomorphic retraction of K onto P. Proof: Clearly, f is continuous since multiplication and u are. If p e Se n E, then f(p) = p since pe = p and u(p) = p. It remains only to show that f is a homomorphism. If x,y c K, then xye e xS n Se = xeS n Se = f(x)S n Sf(y) so that f(xy) = u(xye) = f(x) = f(x)f(y). 3.5.3: Lemma. If x,y e K, then f(x) = f(y) if and only if xS = yS. Proof: If f(x) = f(y), then f(x)S = f(y)S and xS = xeS = f(x)S = f(y)S If xS = yS, then, using the fact that wS n Sp is a subgroup of S for all w,p e K, it is seen that f(x) = xeS n Se n E = xS n Se n E = yS n Se n E = yeS n Se n E = f(y). 3.5.4: Proposition. If K is agreeable with respect to S and if there are at least three minimal right ideals of S, then K satisfies 5.31. 5.31: If x e S, then xt = tS for all t in K or there exists a point t' in K such that xK = t'S. Proof: In the following analytic diagram k is an isomor- phism. S Z(f,S) A U k f K > P Since k(P) is agreeable with respect to S, it fol- lows from Lemma 3.5.1 that k(P) is agreeable with respect to Z(f,S). Corollary 3.4.10 is applicable since k(P) is 2-simple. Thus k(P)(k(P)[-l]) = Z(f,S) since k(P) is an ideal of Z(f,S). Therefore, it follows from Corollary 3.4.10 that if x e S, then c(x)c(t) = cp(t) for all t in K or there exists a point t' in K such that p(x)p(t) = c(t') for all t in K. If c(x)p(t) = c(t), then cp(xt) = cp(t) so that xtS = tS. If c(x)c(t) = p(t') for all t e K, where t' is a fixed point in K, then cp(xt) = cp(t') so that xtS = t'S. This means that xK = x(u(tSIt e K))= t'S. The following proposition may be proved in a manner dual to the proof of the previous proposition. 3.5.5: Proposition. If K is agreeable with respect to S and if there are at least three minimal left ideals of S, then K satisfies 5.32. 5.32: If x e S, then Stx = St for all t in K or there is a point t' in K such that Kx = St'. Next, it is shown that if K satisfies 5.31 and 5.32, then K is agreeable with respect to S. To demonstrate this fact, we shall verify that an arbitrary epimorphism f defined on K satisfied 3.1.9; this verification is done in the next few lemmas. If e e E, then there exists a subgroup H which is maximal with respect to containing e [10]. If e e E n K, then He = es n Se [10]. These two theorems are used ex- tensively in the remainder of this section. 3.5.6: Lemma. If f(k) = f(k'), then f(u(k)) = f(u(k')). Proof: Since f[H u(k) and f[H u(k) are subgroups of f[K], there exists y,y' T such that f(k)y = yf(k) = f(u(k)) and f(k')y' = y'f(k') = f(u(k')). Thus, using the fact that u(k) and u(k') act as units on Hu(k) and Hu(k'), respect- ively, it is seen that f(u(k)) = f(k)y = f(k')y = f(u(k')) f(k')y = f(u(k'))f(k)f(y) = y'f(k')f(u(k)) = y'f(k)f(u(k)) = y'f(k) = y'f(k') = f(u(k')). 3.5.7: Lemma. Let k,k' e K such that f(k) = f(k') and let x e S. (i) If f(u(xu(k))) = f(u(xu(k'))), then f(xk) = f(xk'). (ii) If f(u(u(k)x)) = f(u(u(k)x)), then f(kx)= f(h'x). Proof: (i) Let e' = u(k), e" = u(k'), e = u(xe'), and e2 = u(xe"). Using Lemma 3.5.6, it is seen that f(xe') = f(elxe') = f(e )f(xe) = f(e2)f(xe') = f(e2x) f(e') = f(e2x)f(e") = f(e2xe") = f(xe") so that f(xk) = f(xk) = f(xe'k) f(xe')f(k) f(xe")f(k') = f(xe"k') = f(xk'). The proof of (ii) follows by an argument similar to the above argument. 3.5.8: Lemma. Let e', e", el, e2 e K n E such that f(e') = f(e"). (i) If elS = e2S, Se' = Sel, and Se" = Se2, then f(el) = f(e2). (ii) If Se1 = Se2, e'S = elS, and Se" = Se2, then f(e ) = f(e2). Proof: (i) Since e2, e" C Se" n E, e2e" e Se" n E n e2S = Se2 n E n e2S = (e21. Using the homomorphic properties of f and the hypotheses, it is seen that f(e2) = f(e2e") = f(e2)f(e") = f(e2)f(e') = f(e2e') c f(e2S r Se') = f(e1 n Se1) = f[He ]. Therefore f(e ) = f(e2), since f(e2) is an idempotent and f[Hel] is a group. The proof of (ii) follows from a similar argument. 3.5.9: Lemma. Let p,k c K and let x e S. (i) If Skx = Sp, then Su(k)x = Sp. (ii) If xkS = pS, then xu(k)S = pS. The proof of this lemma is obvious. 3.5.10: Lemma. Let k,k' e K such that f(k) = f(k') and let x e S. (i) If xkS = kS and xk'S = k'S, then f(xk') = f(xk). (ii) If Skx = Sk and Sk'x = Sk', then f(k'x) = f(kx). Proof: (i) Let e' = u(k), e" = u(k'), el = u(xe'), and e2 = u(xe"). It follows that el = u(xe') = e', since xe' e Se' n xe'S = Se' n xks = Se' f kS = Se' r e'S. Simi- larly, e2 = e". Thus, from Lemma 3.5.6, f(el) = f(e2) and hence f(xk) = f(xk') by Lemma 3.5.7. The proof of (ii) is similar. 3.5.11: Lemma. Let k,k' e K such that f(k) = f(k') and let x e S. (i) If xkS = xk'S = t'S for some t' in K, then f(xk) = f(xk'). (ii) If Skx = Sk'x = St" for some t" in K, then f(k'x) = f(kx). Proof: (i) Let e' = u(k), e" = u(k'), e = u(xe'), and e2 = u(xe"). It follows that Se1 = Se' and elS = t'S since xe' e Se' rl xe'S = Se' n t'S. Similarly, Se2 = Se" and e2S = t'S. Therefore, from Lemma 3.5.6, f(e') = f(e") and since eS = u(xe')S = xe'S = xkS = t'S = xk'S = xe"S = u(xe")S = e2S and elS = t'S = e2S, the hy- potheses of Lemma 3.5.8 are satisfied. Thus f(el) = f(e2), and hence, from Lemma 3.5.7, f(xk) = f(xk'). The proof of (ii) follows by a similar argument. 3.5.12: Proposition. If K satisfies 5.31 and 5.32, then K is agreeable with respect to S. Proof: This proposition follows immediately from Lemmas 3.5.10 and 3.5.11. 3.5.13: Lemma. (i) If S contains two minimal right ideals and if f(k) = f(k'), then f(xk) = f(xk') for all x c S. (ii) If S contains two minimal left ideals and if f(k) = f(k'), then f(kx) = f(k'x) for x e S. Proof: (i) There are four cases to be consid- ered: (1) xkS = kS, xk'S = k'S; (2) xkS = ks, xk'S = ks; (3) xkS = k'S, xk'S = k'S; and (4) xkS = k'S, xk'S = kS. (1) follows from Lemma 3.5.10 (i). (2) and (3) follow from Lemma 3.5.11 (ii). (4) Let e' = u(k), e" = u(k'), g = u(xe'), and e2 = u(xe"). It follows that g = u(e"e') since xe' e Se' n xe'S = Se' n k'S = Se' n e"S. Using Lemma 3.5.6 and the idempotentcy property of e", it is seen that f(e") = f(e")f(e") = f(e")f(e') = f(e"e') e f[H ]. But f(e") is also an idempotent so that f(e") = f(g). Similarly, f(e') = f(e2) so that f(g) = f(e") = f(e') = f(e2). Thus, the conclusion follows from Lemma 3.5.7 and (i) holds in all four cases. The proof of (ii) is similar. 3.5.14: Proposition. (i) If S contains two minimal right ideals and two minimal left ideals, then K is agreeable with respect to S. (ii) If S contains two minimal right ideals and if S and K satisfy 5.32, then K is agreeable with respect to S. (iii) If S contains two minimal left ideals and if K satisfies 5.31, then K is agreeable with respect to S. Proof: (i) follows from 3.5.13. (ii) follows from 3.5.13 (i), 3.5.10 (ii), and 3.5.11 (ii). (iii) follows from 3.5.13 (ii), 3.5.10 (i), and 3.5.11 (i). The following theorem combines Proposition 3.5.5, 3.5.6, 3.5.12, and 3.5.14. 3.5.15: Theorem. If K is the minimal ideal of a compact semigroup S, then K is agreeable with respect to S if and only if at least one of the following statements hold: (1) 5.31 and 5.32 are satisfied by K and S. (2) There are exactly two minimal left ideals of S and 5.31 is satisfied by S and K. (3) There are exactly two minimal right ideals of S and 5.32 is satisfied by S and K. (4) There are exactly two minimal left ideals and exactly two minimal right ideals of S. 6. Congruences. Let S be a compact semigroup, let A be a closed sub- semigroup of S, let E be the set of all closed congruences of A, and let i be the set of all epimorphisms defined on A. Recall that there exists a one-to-one correspondence between a and 3. In the previous two sections a subset U of ri was chosen in order to prove the necessity of the two theorems in those sections. In a certain sense, each element of e was minimal in 3. The fact that it is enough to look at a rather than 3 is not accidental. It is the purpose of this section to explain why this is true. However, it is simpler to look at Z rather than a for this purpose. Because of the natural correspondence between 3 and S, the results of this section can be readily interpreted for 3. First, a few preliminary remarks concerning congru- ences need to be made. 3.6.1: Lemma. Let J be a collection of closed congruences of A. Then B' = n (BIB e II is a closed congruence of A. Proof: It is clear that B' is a closed equivalence rela- tion on A. It remains to show that (AA) B' c B' B'(AA). If x c A and (b,b') e B for all B e 91, then (xb,xb') e B for all B e S and (AA) B' c B'. Similarly, B' contains B'(AA). 3.6.2: Lemma. If x,y e A, then there exists a unique min- imal closed congruence of A containing (x,y). This con- gruence will be denoted by F(x,y). Proof: Let F(x,y) be the intersection of all closed con- gruences containing (x,y). 3.6.3: Theorem. If A is a closed subsemigroup of the com- pact semigroup S, then A is agreeable with respect to S if and only if F(a,a') U (AS) is a congruence of S for all a,a' e A. Proof: The necessity follows immediately from Remark 3.1.14. Sufficiency. Suppose B is a congruence of A. It is sufficient, according to Remark 3.1.14, to show that B U (AS) is a congruence of S, i.e., if x e S and (w,z) e B U (AS), then (xw,xz), (wx, zx) e B U (AS). This is clear if (w,z) e AS. If (w,z) e B, then (xw,xz), (wx,zx) C F(w,z) U (AS) since F(w,z) U (AS) is a congruence of S. It follows that (xw,xz), (wx,zx) e B U (AS) since F(w,z) is a subset of B. CHAPTER IV HOMOMORPHIC RETRACTS IN SEMIGROUPS In the previous chapter the concept of homomorphic retraction was introduced to study Borsuk's Paste Job in semigroups. The question of when is a subsemigroup a homo- morphic retract arises because of the role that homomorphic retractions play in the hypothesis of Proposition 3.2.2. This question is studied in the present chapter. A set of necessary and sufficient conditions for the minimal ideal to be a homomorphic retract is given in Section 2 and it is shown how these conditions are applicable to subsemi- groups other than the minimal ideal. A theorem which gives another set of necessary and sufficient conditions for the minimal ideal to be a homomorphic retract is pre- sented in Section 3. 1: Preliminaries and Homomorphic Extensions. Notation: In this section, S is a semigroup and A is a subsemigroup of S. If A is agreeable with respect to S, then Z(f,S) is a "natural" semigroup for all continuous epimorphisms f which are defined on A. If f(A) is identified with its iseomorphic image in Z(f,S), then the natural projection of S onto Z(f,S) may be thought of as a continuous homo- morphic extension of f. Thus, if the special features of Z(f,S) are disregarded except for the fact that f(A) can be iseomorphically embedded in Z(f,S), then the previous chap- ter can be regarded as an attempt to construct a semigroup Z containing f(A) as a subsemigroup such that f has a con- tinuous homomorphic extension to S. In this chapter, the additional restriction that Z = f(A) is made. The follow- ing theorem shows that this problem is equivalent to study- ing the problem of determining when a subsemigroup is a homomorphic retract. 4.1.1: Theorem. Every continuous epimorphism defined on A has a homomorphic extension to S if and only if A is a homomorphic retract of S. Proof: Assume every epimorphism defined on A has a homo- morphic extension to S. In particular, the identity func- tion from A onto A has a homomorphic extension r to S, and this function r is a homomorphic retraction of S onto A. Assume A is a homomorphic retract of S and let f be a continuous epimorphism defined on A. If r denotes a homomorphic retraction of S onto A, then rf is a homomor- phic extension of f to S. Along this same line, it is easily shown that every continuous homomorphism defined on A has a continuous (not necessarily homomorphic) extension to S if and only if A is a retract of S. Thus every continuous function defined on A has a continuous extension to S if and only if every continuous homomorphism defined on A has a continuous ex- tension to S. The following remark is similar. 4.1.2: Remark. The following are equivalent: (1) There exists a continuous function f from S onto A such that the restriction of f to A is a homomorphism.* (2) There exists a continuous epimorphism h from A onto A with the property that if g is a continuous homomor- phism defined on A, then there exists a continuous function k from S to g(A) such that the following diagram is analy- tic. k S g(A) U h 19 h A A Proof: (1) implies (2). Let h be the restriction of f to A. If g is a continuous homomorphism defined on A, then k = gf has the required properties. (2) implies (1). If g is the identity from A onto A, then there exists a continuous function f such that the following diagram is analytic. f S > g(A) U h1 9 h A A *This condition was suggested by Professor W. L. Strother. Thus f restricted to A is h since f(S) = A and f is con- tinuous. We now turn our attention to the problem of deter- mining when A is a homomorphic retract of S; the following example will demonstrate that the minimal ideal of a com- pact semigroup may be a homomorphic retract of the semi- group without the minimal ideal being a group. 4.1.3: Example. Let G be a non-trivial group and let X = Y = Z = [0,1]. Define xy = x for all x,y e X, and de- fine xy = y for all x,y e Y. Let Z have the usual multi- plication. Let S = X x Y x Z x G with co-ordinatewise mul- tiplication. Clearly, the minimal ideal is X x Y x (0) x G. Define r from S to K by r(x,y,z,g) = (x,y,0,g). Since r[(x,y,z,g)(x',y',z',g')] = r(x,y',zz',gg') = (x,y',0,gg') = (x,y,0,g)(x',y',0,g') = r(x,y,z,g)r(x',y',x',g'), r is a homomorphic retraction. The remainder of this section contains two propo- sitions which will be used later. 4.1.4: Proposition. A retraction f of a semigroup S on- to a subset B is a homomorphic retraction if and only if B is a subsemigroup of S and f(xy) = f(f(x)f(y)) for all x,y e S. Proof: The necessity of this condition is obvious. If B is a subsemigroup of S and if f(xy) = f(f(x)f(y)) for all x,y e S, then f(xy) = f(x)f(y) since f(x)f(y) e B. Thus f is a homomorphic retraction. 4.1.5: Proposition, Let f be a hoaororphic retraction of S onto T and let R be a congruence of S such that f(xR) = f(x)R for all x e S. Then there exists a homomorphic re- traction f* of S/R onto c(T) where cp is the natural pro- jection of S onto S/R. Moreover, the following diagram is analytic. f* S/R > S/R 1J f I s s Proof: The fact that there exists a continuous homomor- phism f* which makes the above diagram analytic is well known. Therefore, it is sufficient to show that f* is a retraction of S/R onto cp(T). If x c S, then f*cp(x) = cpf(x) c qc(T). If z e p(T), then there exists x e T such that cp(x) = z. Thus f*(z) = f*p(x) = pf(x) = c(x) = z so that f* is a homomorphic re- traction of S/R onto c(T). 2. The First Theorem. In Sections 2 and 3 it is necessary to make use of various structure theorems for the minimal ideal of a semi- group. These may be found in [2], [10], [16], [18], and [20]. In the remainder of this chapter, K is the minimal ideal of S. 4.2.1: Proposition. Let R be a right ideal of a semigroup S and let there exist a continuous function f from S into R such that f has the following properties: (1) f(xy)xy = f(x)x f(y)y for all x,y e S, (2) f(x)x = x for all x e R; then R is a homomorphic retract of S. Proof: The function r from S to R is defined by r(x) = f(x)x for all x e S. It will be shown that the function r is a homomorphic retraction of S onto R. Since multiplication and f are continuous, r is con- tinuous. It follows that r is a retraction because (2) im- plies r(x) = x for all x e R. Also, by (1) it is seen that r(xy) = f(x)xf(y)y = r(x)r(y) for all x,y e S so that r is a homomorphism. 4.2.2: Proposition. Let L be a left ideal of a semigroup S and let there exist a continuous function f from S into L such that f has the following properties: (1) xyf(xy) = xf(x)yf(y) for all x,y e S, (2) xf(x) = x for all x e L; then L is a homomorphic retract of S. The proof of this proposition is dual to the proof of Proposition 4.2.1. 4.2.3: Corollary. Let R be a right ideal of a semigroup S. If R contains a point which acts as a unit for R, then R is a homomorphic retract of S. Proof: Let a e R such that a is a unit for R. Define f from S into R by f(x) = a for all x e S. To establish the corollary it is sufficient to verify that f satisfies the hypotheses of Proposition 4.2.1. Clearly, f is continuous, and using the fact that a is a right unit for the points of R, f(xy)xy = axy = (ax)y = (axa)y = (ax)(ay) = (f(x)x) (f(y)y) for all xy e S. Also, if x c R, then the fact that a is a left unit for the points of R implies that f(x)x = ax = x. 4.2.4: Corollary. Let L be a left ideal of a semigroup S. If L contains a point which acts as a unit for L. then L is a homomorphic retract of S. The proof of this corollary is dual to the proof of Corollary 4.2.3. The next corollary appears in [8, p. 289]. 4.2.5: Corollary. Let I be an ideal of a semigroup S. If I has an identity for I, then I is a homomorphic re- tract of S. The proof of Corollary 4.2.5 is immediate from Corollary 4.2.4. Notation: In the rest of this chapter S is a com- pact semigroup, K is the minimal ideal of S, E denotes the set of idempotents of S, and H = U(Hele e E) where He is the maximal subgroup of S which contains e. 4.2.6: Lemma. Let A be a closed subset of S and let f be a function from S to A with the following properties: (1) f(x)x = xf(x) for all x in S, (2) if y e A and yx = xy for some x in S, then y = f(x). Then f is continuous. Proof:* Let m be the function defined from S x S to S x S *This proof was suggested by Professor A. D. Wallace. by m(x,y) = (xy,yx). m is clearly continuous so that m- [AS 0 (S X A)] is closed. But m-l[As n (S x A)] = ((x,y): xy = yx and y e A) = ((x,f(x)):x e S) so that f is continuous. 4.2.7: Definition. A function f from S into S is a re- duction of S if and only if f has the following proper- ties: 4.2.7.1 If x C S and if y e f(S), then fOx) = y if and only if xy = yx. 4.2.7.2 If x,y e S, then f(xy)xyf(xy) = f(x)xyf(y). A subset A of S is called a reduct of S if and only if there exists a reduction f of S such that f(S) = A. 4.2.8: Proposition. If the image of a reduction is closed, then the reduction is a retraction. Proof: If f is a reduction, then Lemma 4.2.4 and 4.2.7.1 imply f is continuous and the fact f(a) commutes with f(a) 2 for all a e S implies that f = f; thus f is a retraction. 4.2.9: Proposition. Let R be a closed right ideal of S contained in U (eSele e R n E). If R n E is a reduct of S, then R is a homomorphic retract of S. Proof: Let f denote the reduction of S onto R f E. It is sufficient to verify that f satisfies the hypotheses of Proposition 4.2.1. Lemma 4.2.4 guarantees that f is con- tinuous. Using 4.2.7.1 and 4.2.7.2, it is seen that f(xy)xy = f(xy)f(xy)xy = f(xy)xyf(xy) = f(x)xyf(y) = f(x)xf(y)y for all x,y e S. If x is a point in R, then there exists a point e in R n E such that x C eSe. Thus ex = x = xe so that f(x) = e. 4.2.10: Corollary. Let R be a closed right ideal of S contained in H = U (Hele e E). If R n E is a reduct of S, then R is a homomorphic retract of S. Proof: If x c R, then there exists e e E such that x c He' Since He is a group, there exists a point y e He such that xy = e so that e c R. Thus R is contained in U (eSele c E n R) and it follows from Proposition 4.2.9 that R is a ho- momorphic retract of S. The following two statements are the duals of the previous two statements. 4.2.11: Proposition. Let L be a closed left ideal of S contained in U (esele e L n E). If L n E is a reduct of S, then L is a homomorphic retract of S. 4.2.12: Corollary. Let L be a closed left ideal of S contained in H. If L n E is a reduct of S, then L is a homomorphic retract of S. 4.2.13: Proposition. Let I be an ideal of S which is contained in H. If I is a homomorphic retract of S, then there exists a continuous function f from S into E n I with the properties that f(x)x = xf(x) and that f(xy)xyf(xy) = f(x)xyf(y) for all x and y in S. Proof: Let r be a homomorphic retract of S onto I and de- fine f from S into E n I by f = ur. Since r and u are con- tinuous, f is continuous. Using the definitions of u and f and the homomorphic retraction properties of r, it is seen that xf(x) = r(xf(x)) = r(x)rf(x) = r(x)f(x) = r(x) = f(x)r(x) = rf(x)r(x) = r(f(x)x) = f(x)x for all x in S. Also, f(xy)xyf(xy) = r(f(xy)xy)f(xy) = rf(xy)r(xy)f(xy) = f(xy)r(xy)f(xy) = r(xy) = r(x)r(y) = f(x)r(x)r(y)f(y) = rf(x)r(x)r(y)rf(y) = r(f(x)xyf(y)) = f(x)xyf(y) for all x,y e S. 4.2.14: Example. This is an example of a semigroup S with an ideal I which is contained in H and which is a homomorphic retract of S, but I n E is not a reduct of S. Let S be the unit interval with the multiplication given by xy = min(x,y) and let I = [0,J]. Since I n E = I and S is Abelian, no function from S onto I n E will sat- isfy 4.2.7.1. 4.2.15: Theorem. The minimal ideal K is a homomorphic retract of S if and only if K n E is a reduct of S. Proof: The necessity follows from Corollary 4.2.7. Define f from S to K n E by f = ur where r is a homomorphic retraction of S onto K. It follows from Pro- position 4.2.9 that f satisfies 4.2.5.1, 4.2.5.3, and 4.2.5.4. It remains only to verify 4.2.5.2. Suppose g is a function from S onto K n E such that g(x)x = xg(x) for all x in S. Let x be a fixed point in S. It must be shown g(x) = f(x). Since f = ur, r(x).vr(x) f(x) = vr(x)-r(x). Thus g(x)f(x) = g(x)r(x)vr(x) = r(g(x)x)vr(x) = r(xg(x))vr(x) = r(x)g(x)vr(x) e f(x)S n g(x)S so that f(x)S = g(x)S. In a similar fashion it may be shown that Sf(x) = Sg(x). Therefore, (g(x)) = g(x)S n Sg(x) n E = f(x)S n Sf(x) n E = (f(x)l. It is instructive to look at the case when S is simple, i.e., S = K. Clearly, K is a homomorphic retract of S so that there exists a reduction of S onto E. This reduction is u and it is clear that u(x) = x = xu(x) for all x c S. If e e E and ex = xe, then ex c eS n u(x)S n Su(x) n Se. Thus (e) = eS 0 E n Se = u(x)S 0 E n Su(x) = (u(x)]. The next corollary says that if in an arbitrary semigroup S there exists a reduction of S onto K n E, then this reduction is an extension of u. 4,2.16: Corollary. If I is a closed ideal contained in H and if f is a reduction of S onto I n E, then I is a homomorphic retract of S, and if r is a homomorphic re- traction of S onto I, then f = ur. Thus, there is only one homomorphic retraction of S onto I. Proof: The fact that I is a homomorphic retract of S is exactly the statement of Proposition 4.2.9. If x e S, then, using the homomorphic retraction properties of r, it is seen that xur(x) = r(xur(x)) = r(x)rur(x) = r(x)ur(x) = r(x) = ur(x)r(x) = rur(x)r(x) r(ur(x)x) = ur(x)x. Thus it follows from 4.2.7.2 that f(x) = ur(x) for all x e S. It remains to show that there is only one homomor- phic retraction of S onto I. This is done by showing that if r is a homomorphic retraction of S onto I, then r(x) = f(x)x. Using the definition of u and the fact that f = ur, it is seen that r(x) = ur(x)r(x) = f(x)r(x) = rf(x)r(x) = r(f(x)x) = f(x)x. 4.2.17: Corollary. If f is a reduction of S onto K n E, then f(xy) = f(f(x)f(y)) for all x,y e S. Proof: If x,y e S, then f(xy)xyf(xy) = f(x)xyf(y) c f(xy)Sf(xy) n f(x)Sf(y). Thus Hu(f(x)f(y)) = Hf(xy) so that u(f(x)f(y)) = f(xy). But u(f(x)f(y)) = ur(f(x)f(y)) = f(f(x)f(y)) where r is any homomorphic retraction of S onto K. 4.2.18: Corollary. If K is a homomorphic retract of S and if K n E is a subsemigroup of S, then K 0 E is a sub- semigroup of S. The proof of this corollary is immediate from Pro- positions 4.1.4, 4.2.8, and Corollary 4.2.17. 4.2.19: Corollary. Let S be a compact semigroup with unit. The minimal ideal of S is a homomorphic retract of S if and only if the minimal ideal is a group. Proof: The sufficiency follows immediately from Corollary 4.2.5. Necessity. Let 1 denote the unit of S. If e,e' e K, then le = e = el and le' = e' = e'l. But if K is a homo- morphic retract of S, then K n E is a reduct of S so that e = e' and hence the cardinality of K n E is one. Thus K is a group. 4.2.20: Example. This is an example of a semigroup S con- taining a homomorphic retract which contains no reduct of S. Let S be a simple semigroup which contains two dis- tinct minimal left ideals, L and L'. It was shown in Lem- ma 3.5.2 that L n E is a homomorphic retract of S. It remains only to show that L n E contains no reduct of S. Assume the contrary, i.e., L n E does contain a reduct G of S. Since L' n E is not empty, it contains a point p. Thus, there exists a point g in G such that pg = gp. Since pg e L and gp e L', then L' n L is not empty and so L = L'. This is a contradiction. Therefore, G does not exist. 3. The Second Theorem. Theorem 3.5.15 gives a set of necessary and suffi- cient conditions for K to be agreeable with respect to S. Since Proposition 3.2.2 says that if K is a homomorphic retract of S, then K is agreeable with respect to S, it must be the case that a subset of the conditions in 3.5.15 are necessary and sufficient for K to be a homomorphic re- tract of S. Such a theorem is proved in this section. 4.3.1: Lemma. Let g and g' be functions from S into K. The following are equivalent: (1) g(x)S = xkS and Skx = Sg'(x) for all x e S and k e K. (2) xK = g(x)S and Sg'(x) = Kx for all x e S. Proof: (1) implies (2). Clearly, xK is a subset of U (xkS k e K) = g(x)S. Since xK is a right ideal and g(x)S is a minimal right ideal, xK = g(x)S. A dual argument dem- onstrates that Kx = Sg'(x). (2) implies (1). If x e S and k e K, then xk e g(x)S n xkS so that xkS = G(x)S. A dual argument shows that Skx = Sg'(x). 4.3.2: Theorem. The following are equivalent: (1) K is a homomorphic retract of S. (2) There exist functions g and g' from S into K such that xK = g(x)S and Kx = Sg(x) for all x e S. Proof: (1) implies (2). Since K is a homomorphic retract of S, there exists a reduction f of S onto K n E. Using the reduction properties of f and Corollary 4.2.16, it is seen that xkS = xkf(xk)S = f(xk)xkS = f(xk)S = f(f(x)f(k))S = u(f(x)f(k))S = f(x)f(k)S = f(x)S for all x e S and k e K. In a similar fashion, it is seen that Skx = Sf(x) for all x e S and k e K. Let g = g' = f. (2) follows from Lemma 4.3.1. (2) implies (1). It is enough to prove the function f defined by f(x) = u(g(x)g'(x)) is a reduction of S onto K n E. Clearly, f is a function from S into K f E, so that it is only necessary to show that f satisfies 4.2.7.1 and 4.2.7.2. Since xf(x) e xf(x)S = g(x)S = g(x)g'(x)S = u(g(x)g'(x))S = f(x)S, it follows that xf(x) = f(x)xf(x). Using a dual argument, it may be seen that f(x)x = f(x)xf(x) and hence f(x)x = xf(x). If e e K n E, if x e S, and if xe = ex, then ex e Sex n Se = Sg'(x) so that Se = Sg'(x). Similarly, eS = g(x)S. Hence, g(x)g'(x) e eS n Se and thus f(x) = u(g(x)g'(x)) = e. It remains only to show that f(xy)xyf(xy) = f(x)xyf(y) for all x,y e S. First it is shown that f(xy)f(x) = f(x) and f(y)f(xy) = f(y). Since g(xy)S = xyf(x)S = g(x)S and Sg'(xy) = Sf(x)xy = Sg'(x), f(xy) e g(x)S n Sg'(y) = f(x)S n Sf(y). Thus f(xy)S = f(x)S and Sf(xy) = Sf(y) and hence f(xy)f(x) = f(x) and f(y)f(xy) f(y) for all x,y e S. Next, since f(xy)x C Sf(xy) = Sg'(x) = Sf(x), it follows that f(xy)x = f(xy)xf(x) = f(xy)f(x)x = f(x)x. In a similar fashion, it follows that yf(xy) = yf(y). Thus f(xy)xyf(xy) = f(x)xyf(y). 4.3.3: Corollary. The following are equivalent: (1) K n E is a reduct of S. (2) There exists a function f from S to K n E such that xK = f(x)S and Kx = Sf(x) for all x e S. Proof: (1) implies (2). It was shown in the proof of Theorem 4.3.2 that a reduction of S onto K n E will satis- fy (2). (2) implies (1). (2) implies K is a homomorphic retract of S which implies K n E is a reduct of S. CHAPTER V APPLICATION OF HOMOMORPHIC RETRACTIONS TO RELATIVE IDEALS Let S be a semigroup and let A and T be non-empty subsets of S. Then A is a T-ideal if and only if TA U AT is a subset of A, A is right T-ideal if and only if AT is a subset of A, and A is a left T-ideal if and only if TA is a subset of A. A T-ideal is also known as a relative ideal and they have been studied in [17], [18], and [19]. The main problem studied in this chapter is to find conditions which will insure that a minimal T-ideal is a retract of S. The standing hypothesis in Sections 3 and 4 is that T is a homomorphic retract of S. In Section 3 there are propositions which say that if T is a homomorphic retract of S and if one of five other conditions are satis- fied, then a minimal T-ideal is a retract of S. In section 4 it is shown that if L and R are minimal left and right T-ideals, respectively, and if both L and R are subsemi- groups, then the minimal T-ideal LR is a retract of S in case T is a homomorphic retract of S. In Section 1 various examples of minimal T-ideals are presented and Section 2 contains some technical propo- sitions which are used in Sections 3 and 4. 1: Examples of Minimal T-ideals. 5.1.1: Example. Let S = [-1,1] under the usual multipli- cation and let T = (-1,1). Then (-1,1l, (0), and (-J,~) are minimal T-ideals and the second is a retract of S. 5.1.2: Example. Let X = [0,1] under the multiplication xy = x, let P = {x 0:0 ; x 1 U {[O0 :0 & y- 1l, let S = X x P, and let Q = {[1, [1 0], [0, [ 0]]}. Then {[1, [,b ]0 [0, o ]]} is a minimal Q-ideal. If T = x [l 0 then X x{ 0 is a minimal T-ideal which is a retract of S. 5.1.3: Example. Let X and P be as in t.1.2, let Y = Z = [0,1] with the multiplication in Y given by xy = y, and let Z have the usual multiplication. Let S = P x X X G x Y x Z where G is any compact topological group. If T = 0{[ x X X X G x Y x (1i, then any minimal T-ideal is a retract of S. 5.1.4: Example. Let X and Y be as in Example 5.1.3, let Z = [0,1] with the multiplication xy = min(x,y), and let S = X x Z x Y with co-ordinatewise multiplication. If T = X x (1) x Y, then the collection of minimal T-ideals is (X x (z) x Yjz e Z); each element of this collection is a retract of S. The collections of minimal left and right T-ideals are (X x (z) x (y)jz c Z, y e Y) and I(x) x [z} X Ylx c X, z c Z), respectively; each element of these two collections is a subsemigroup of S. 5.1.5: Example. Let S = E2 with the usual complex number multiplication. If T = S then T is a minimal T-ideal; if Q = (-1,1), then [-3,w) is a minimal Q-ideal but neither T nor i-s,}] is a retract of S. 5.1.6: Example. Let X = S1 with multiplication xy = x, let Z = [0,1] with the usual multiplication, and let S = (X x Z)/(X x (0o). If T = ((x,l)10 arg x < -], then ((x,j)|0 < arg x e I- is a minimal T-ideal and an absolute retract. 5.1.7: Example. This example shows that there exists a semigroup S which contains a homomorphic retract T and a minimal T-ideal I which is a retract of S, but I is not a homomorphic retract of S. Moreover, I = LR where L and R are minimal left and right T-ideals which are intersecting subsemigroups of S. Let P = {V0 Ilx is a real number and 0 t x < 1 U {0 f]ly is a real number and 0 y 1}, let X be [0,1] with the usual multiplication, let Y be [0,1] with the mul- tiplication given by xy = min(x,y), and let S = X x P x Y with co-ordinatewise multiplication. If T = X X P x (l), then T is a homomorphic retract of S and I = X x K x {Ch is a minimal T-ideal where K= 0[0 Y lY is a real number and 0 y & . To see that I is not a homomorphic retract of S, we assume the contrary, i.e., there exists a homomorphic re- traction f of S onto I. Using the homomorphic retraction properties of f, it is seen that (1, 0 = O 1[ (, [o~J], 1)=<, [o], o]^ [o 1] ,) f(1, [ ], [o l, ). Since it is of the form (x,k, ) where x e X and k c K. Therefore (x,k, ) (1, [0 fl, ) = (x,k,j); thus, f(l, =) = 1, ). Similarly, it is seen that (1, = 1 [01 1) (l, [0 ', 1) 2: Preliminary Propositions. Throughout the rest of this chapter, S is a compact semigroup, T is a non-empty subset of S, and f is a homo- morphic retraction of S onto T. It should be noted that this implies T is a closed subsemigroup of S. The minimal ideal of T is denoted by K, H is the union of all subgroups of S, E is the set of all idempotents of S, u is the func- tion from H onto E defined by "u(x) is the unit of a group which contains x" [20], and v is the function from H onto H defined by "v(x) is the inverse of x in a group which con- tains x" [20]. 5.2.1: Lemma. If I is a minimal T-ideal, then f(I) = K. Proof: It follows from the Homomorphism Theorem [15] that f(I) is an ideal of T and so K is a subset of f(I). Since K n E is not empty, there exists a point a in I such that f(a) s E n K. The fact that Tf(a)af(a)T is a T-ideal con- tained in I implies that Tf(a)af(a)T = I. Thus, using the homomorphic retraction properties of f, it is seen that f(l) = f(Tf(a)af(a)T) = f(T)f2(a)f(a)f2(a)f(T) = Tf(a)f(a)f(a)T = Tf(a)T = K. 5.2.2: Proposition. If I is a minimal T-ideal, then K n E is a reduct of K U(U(t"In = 1,2,...]). Moreover, if r is the reduction of K (UU(Iln = 1,2,... ) onto K n E, then r(x)x = xr(x) = x for all x e K U(UItn n = 1,2,...)). Proof: Let r = uf; it must be shown that r has the follow- ing properties. (1) r(x)x = xr(x) = x for all x e K U(U(In n = 1,2, (2) If e e K n E and ex = xe, then r(x) = e. (3) If x,y e K U(U(In n = 1,2,...)), then r(xy) xyr(xy) = r(x)xyr(y). It should be noted that (3) follows immediately from (1). If x e K, then (1) and (2) are clear. Suppose x e U(Inln = 1,2,...]. Then x = x x2,9...xn, where xi I for i = 1,2,3,...n. From Lemma 5.2.1, it is seen that there exists a minimal left ideal L of T and a minimal right o ideal R of T such that f(x) e R 0n L I = LoX R there 0 0 0 0 1 o exist tl, t e L and p,, pn e R such that tlxlPl = x and tn xPn = xn. Thus f(x) = tl f(x lPlX2".tnx p e (Lo n Ro) n (tlS n Sp ) so that tlS = R and Sp = L It follows that tlPn e Lo n Ro and so [uf(x)]x = uf(x)tlx lpx2... xn-ltnx = txlp ...xn- xpn = x = x[uf(x)]. 5.2.3: Corollary. If a s K U(U(In n = 1,2,3,...] and f(a) e E, then f(a) e f(a)Sf(a) and f(a)a = af(a) = a. Proof: If r is defined as in Proposition 5.2.2, then r(a) = uf(a) = f(a) and the corollary follows from Proposition 5.2.2. 5.2.4: Lemma. If L is a minimal left T-ideal, then f(L) is a minimal left ideal of T. Proof: Since f(L) is a left ideal of T, it contains a min- imal left ideal L of T. Because L is not empty, there exists a point a in L such that f(a) c L Thus L = Tf(a) = f(T)f(a) = f(Ta) = f(L). 5.2.5: Lemma. If R is a minimal right T-ideal, then f(R) is a minimal right ideal of T. The proof is dual to that of 5.2.4. 5.2.6: Lemma. Let L and R be minimal left and minimal o o right ideals of T, respectively. If t, e Lo, if t2 c Ro' if u(Lo n RO) = e, and if t t2 = e, then t = v(t2). Proof: Since t t2 = e e Lo n Ro n tT n Tt2, then tl tIT = R and t2 e Tt2 = L Thus tl and t2 are in L n R and t = v(t). o o 1 t2) 5.2.7: Lemma. Let I be a minimal T-ideal and let a e I such that f(a) e E. If x e I and f(x) = f(aj), then there exists a point t e K such that x = txv(t). Moreover, if x = t1at2 where tl e Tf(a) and t2 e f(a)T, then tl = v(t2). Proof: Since I = Tf(a)af(a)T, there exist points tL e Tf(a) and t2 e f(a)T such that x = t1at2. Thus f(a) = f(x) = f(tlat2) = tlf(a)t2 = tlt2 and so it follows from Lemma 5.2.5 that tl = v(t2). 5.2.8: Proposition. If L is a minimal left T-ideal, then f restricted to L is a one-to-one function. Proof: Since f(L) is a minimal left ideal of T, there exists a e L such that f(a) is idempotent. If x,y e L, then f(L)a = L implies that there exist tlt2 e f(L) such that x = t1a and y = t2a. If f(x) = f(y), then tl = tlf(a) = f(ta) = f(x) = f(y) = f(t2a) = t2f(a) = t2 because f is a homomorphic retraction. 5.2.9: Proposition. If R is a minimal right T-ideal, then f restricted to R is a one-to-one function. The proof is dual to the proof of Proposition 5.2.7. 5.2.10: Corollary. If L is a minimal left T-ideal, if R is a minimal right T-ideal, and if L n R is not empty, then f restricted to R is a one-to-one function. The proof is clear. 5.2.11: Proposition. If L and L' are minimal left T- ideals and if f(L) n f(L') is not empty, then f(L) = f(L'). Proof: This follows immediately from the fact that f(L) and f(L') are minimal left ideals of T. The following is the dual statement of the previous proposition. 5.2.12: Proposition. If R and R' are minimal right T- ideals and if f(R) n f(R') is not empty, then f(R) = f(R'). 3: Various Sufficient Conditions. In this section, it is shown that if T is a homo- morphic retract of S, if I is a minimal T-ideal of S, and if one of the following five conditions holds, then I is a retract of S: (1) I has a cut point, (2) f restricted to I is one-to-one, (3) K is contained in E, (4) T is normal in S, and (5) H = ((x,y)|x U Tx = y U Ty and y U xT = y U yT1 is a congruence of S. 5.3.1: Proposition. If I is a minimal T-ideal and if f restricted to I is one-to-one, then I is a retract of S. Proof: It has been shown that there exists a retraction -l q of T onto K [20]. The function (flI) qf is a retraction of S onto I since if x e I, then (flI) qf(x) = (flI) f(x) =x. 5.3.2: Corollary. If I is a minimal T-ideal and if K is a subset of E, then I is a retract of S. Proof: It is sufficient to show that f restricted to I is a one-to-one function. Let a e I, let L = Tf(a), and let R = f(a)T, then K = LR and I = LaR. If x,y e I and if f(x) = f(y), then there exist tl, t' L and t2, t2' R such that x = tlat2 and y = t 'at2'. The fact that f(a) is a right unit for L implies that tl 2 = f(tlat2) = f(tl'at2') = tl't2' so that tlT = t1T and Tt2 = Tt21. Because of the hypothesis that K is a subset of E, (tl] = t1T n Ttl = tl'T D Ta = tl'T n Ttl = It 1'. In a similar fashion, it is seen that t2 t2' and hence x = tlat2 = tI at2 = y; thus f restricted to I is one-to-one. 5.3.3: Theorem. If I is a minimal T-ideal and if there is a point a in I such that Ta is a subset of aT, then I is a minimal left T-ideal. Proof: Since I is the union of minimal left T-ideals [15], there exists a minimal left T-ideal L such that a e L. Thus L = Ta and I = TaT c TTa c Ta = L. But L is a subset of I so that L = I. 5.3.4: Theorem. If I is a minimal T-ideal and if there exists a e I such that aT is a subset of Ta, then I is min- imal right T-ideal. The proof is dual to that of Theorem 5.3.3. 5.3.5: Corollary. If I is a minimal T-ideal and if T is normal in I, i.e., xT = Tx for all x e I, then I is a min- imal left and a minimal right T-ideal. It should be noted that the only assumption about T in 5.3.3, 5.3.4, and 5.3.5 is that T is a closed subsemi- group of S. 5.3.6: Corollary. If I satisfies the hypotheses of 5.3.3 or of 5.3.4, then I is a retract of S. Proof: By 5.3.3, I is a minimal left T-ideal and hence by 5.2.8, f restricted to I is one-to-one. Thus, 5.3.1 im- plies that I is a retract of S. 5.3.7: Corollary. If I is a minimal T-ideal and if there exists a point a in I such that Ta = a, then I is a retract of S. Proof: Since I is the union of minimal left T-ideals and the union of minimal right T-ideals [17], there exist a minimal left T-ideal L and a minimal right T-ideal R such that a e R n L. Therefore, L = Ta and R = aT. Thus Ta = (a) c aT and the hypotheses of 5.3.3 are satisfied so that it follows from Corollary 5.3.6 that I is a retract of S. 5.3.8: Corollary. If I is a minimal T-ideal and if there exists a point a in I such that aT = a, then I is a retract of S. The proof is dual to that of Corollary 5.3.7. 5.3.9: Corollary. If I is a minimal T-ideal and if I has a cut point, then I is a retract of S. Proof: It follows from Faucett's Theorem [15] that aT = A or Ta = a for all a in I. Thus the hypotheses of 5.3.7 or of 5.3.8 are satisfied and the conclusion follows: Recall that L(x) = x U Tx, R(x) = x U xT, L (ylL(x) = L(y)), Rx = (ylR(x) = R(y), Hx = L n R M = ((x,y)|x e H ), and H is a closed equivalence relation on S [18]. In the rest of this section it is assumed that U is a congruence of S. Notation: Let S* = S/H, let p be the projection of S onto S*, let T* = cp(T), let I* = cp(I), and let K* = cp(K). It follows from Proposition 4.1.5 that there exists a homomorphic retraction f* of S* onto T* such that the fol- lowing diagram commutes. f* f S S 5.3.10: Lemma. If I is a minimal I-ideal, then I* is a minimal T*-ideal. Proof: Let J* be a minimal T*-ideal contained in I* and let a e I such that p(a) e J*. Then J* = T*cp(a)T* = cp(T)cp(a)C(T) = cp(TaT) = cp(I) = I*. It should be noted that K* is the minimal ideal of T* and every point in K* is an idempotent. 5.3.11: Theorem. If # is a congruence of S, the I is a retract of S. Proof: It is sufficient to show that f restricted to I is one-to-one. If x,y e I and f(x) = f(y), then f*p(x) = f*p(y). But cp(x) = c(y) by Corollary 5.3.4. Thus y e H c R = x x Tx since x is an element of a minimal right T-ideal. Therefore, f restricted to Tx one-to-one implies that x = y. 4: The Subsemiqroup Case. In this section, L and R are minimal left and right T-ideals, respectively, both of which are subsemigroups of S and such that R n L is not empty. In [15] it is proved that I = LR is a minimal T-ideal. In this section, it is shown that I is a retract of S. Since R n L is not empty, it is a group and has a unique idempotent a. If f(R) = Ro and f(L) = Lo, then L L a, R = aR and I = L aR = LaR = LR. o o o o In the following two propositions the hypothesis that T is a homomorphic retract of S may be replaced with the hypothesis that T is a closed subsemigroup of S. 5.4.1: Proposition. If L' is a minimal left T-ideal con- tained in I, then L' is a subsemigroup of S. Proof: If x L' c I = L aR then there exist t, e L and o 1 o t e R such that x = tlat2. Since L' = Lx = L tlat = 2 o 1 2 o o l 2 Loat2 = Lt2 and the left T-ideal L is a subsemigroup, it follows that L'L' = Lt2Lt2 c LLt2 c Lt2 = L'. 5.4.2: Proposition. If R' is a minimal right T-ideal con- tained in I, then R' is a subsemigroup of S. The proof of this proposition is dual to that of Pro- position 5.4.1. 5.4.3: Lemma. If x e I and if f(x) = f(a), then x = a. Proof: Since I = L aR there exist t eL Lo and t2 e R such that x = tlat2. It follows from Lemma 5.2.6 that t = (t2). Using [15, Proposition 20B] it is seen that tla = taf(a) e R aL = (R a)(aL ) = RL = R n L. Since R n L is a group, there exists a unique point k C R n L such that t ak = a; k = at for some t e R since k e R = aR Be- 1 o o cause f is a homomorphic retraction, f(k) = f(at) = f(a)t = t so that k = af(k). Thus f(a) = f(tlak) = tlf(k) and hence f(k) = v(tl) = t2. It follows that x = tlat2 tlaat2 = tlaaf(k) = tlak = a. 5.4.4: Proposition. If L' and L" are minimal left T- ideals and if f(L") n f(L') is not empty, then L" = L'. Proof: If x e L', then there exists a minimal right T- ideal R' contained in I such that x e L' n R' [15, Pro- position 20A]. From Lemmas 5.4.1, 5.4.2, and |