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 counterexamples; 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 Tideals 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
(XA) homeomorphically onto (Z(f,X)k(Y)).
This lemma says that Z(f,X) = J(XA) U k(Y) and
that c(XA) is topologically the same as (XA) 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
AlexanderKolmogoroff 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 coboundary 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 MayerVietorsis 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 onetoone. 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
deyerVietorsis 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 coboundary 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 coboundary operators.
Hl(k(Y)) HP(Z(f,X),k(Y)) HP(Z(f,X))> 0
k*
HPl(y) cp'* cp
f*
6 i*
HP1(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'[HPlk(Y)] = 1'*6f*HP1(Y). Since cp'* is an isomor
phism, HP(Z(f,X),k(Y))/K[i*] is isomorphic to HP(X,A)/
6f*'[HP1( ) .
The exactness of the sequence 0  H (X,A) >
H (Z(f,X)) > 0 implies (ii).
Let En be the ncell and let S nbe the boundary
of En. If the coefficient group is the group of integers
G, then Hn1(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 = Sn1, it follows that
Hn(Z(f,E )) is isomorphic to H (E,Snl)/f*H (S ),
which is also isomorphic to G/(deg f)G.
1.2.11: Lemma. If HP1(X,A) = 0, then HP1(Z(f,X)) is
isomorphic to K[6f*] where 6 is the coboundary operator
from HP1(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 > HPl(Z(f,X)) > HP(k(Y))  HP(Z(f,X),k(Y))
1 k*
HP1y) 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 = Sn1 = A, then Hn1(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 nonto 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 = n1, p = n, or p 0, n1, n.
The following corollary is clear because of these
computations.
1.2.12: Corollary. If f and g are continuous maps from
S nonto S nand 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 PointSet 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'gklkf(x) = k'gklkf(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 onetoone. 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'gklkf(x)
= k"gk kf(y), then cp"(x) = gf(x) = gf(y) = cp"(y) since k'
is onetoone.
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 wellbehaved and Z(g,X) can be even more
pathological (See [5]). However, in case X is the twocell,
E2 and A is the onesphere 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 twocell or twosphere.
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 twocell 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 twosphere, 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 twocell 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 twocell,
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'cl(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'gk1 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: THomotopy.
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 Thomotopic 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 Thomotopic functions. The idea of Thomotopy
las been used effectively in [15].
Some of the usual concepts involving Ihomotopy 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 Thomo
:opic to the identity.
A space X is Tcontractible to a point x from t on
_ if and only if the injection of X onto X is Thomotopic
it t, t' to the constant xvalued 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 (twosided) 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 (twosided) zero if and only if p is a left and
a right zero for m.
The connection between Thomotopy and topological al
gebra is quite strong. If T is a connected space and t,t'
e T, then T is Tcontractible 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 Thomotopy over Ihomotopy. The following ob
servations, which were made by Professor A. D. Wallace,
will serve to illustrate that Thomotopy is a proper gen
eralization of Ihomotopy.
2.0.4: Observation. If X is Tcontractible, X is acyclic.
2.0.5: Observation. If T is a topological semilattice,
then T is Tcontractible.
2.0.6: Observation. If X is Icontractible 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 Lcontract
ible but not Icontractible.
Since the nsphere is not acyclic, not every contin
uum is selfcontractible.
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 Thomotopy and Ihomotopy
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 Thomo
topic to g at t,t', then f is Ihomotopic 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 Ihomotopic 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 Ihomotopic to g, then
f is Thomotopic 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 Ihomotopic, 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 Thomotopic to g.
These two lemmas say that if T is arcwise connected
and normal, then f is Thomotopic to g if and only if f is
Ihomotopic to g.
Some properties of Tdeformation retracts are studied
in the remainder of this section.
2.0.10: Lemma. Let 0, 1 e T. If T is selfcontractible
to 0 from 1 on 0, then T x (0) is a Tdeformation retract of
T x T. Also, (0) x T is a Tdeformation 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 Tdeformation retracts.
2.0.11: Lemma. If A is a Tdeformation 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 Tdeformation 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 Tdeformation retract of X,
then HP(XA) is trivial for all nonnegative 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 Tdeformation retract of X, then
there exists a continuous function f' from X to X such that
f' is Thomotopic 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] = mp'l [B], for if B is closed, then
em 1[B] is closed. If z e k1B], then c'm l(z) =
k(z) e B so that z e em l[B]. Conversely, if
z e em1 C [B], then k(z) = p'm1 (z) e B and the equality
follows.
In order to show that k is a onetoone 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 TMappinq
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
Tcylinder of f over X at 0,1 is the Borsuk's Paste Job
Z(f',o ). Since this partial mapping Tcylinder 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 Tcylinder 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 Tcontract
ible from 1 to 0.
2.2.2: Lemma. If p is a nonnegative 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 Thomotopic 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 Tdefor
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 nonnegative 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 2l (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) = lv212 c2(a,l) = cl(a) = k.f(a) = klk212(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 TCylinders.
The purpose of this section is purely technical. A
relation between two partial mapping Tcylinders 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 onetoone
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 Thomo
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 Thomotopic 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 nonnegative
integers p.
Proof: Since f is Thomotopic 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 restated 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
nonnegative 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 Thomotopic 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 nonempty Hausdorff space together
with a continuous, associative multiplication. Precisely,
a semigroup is a nonempty Hausdorff space S and a function
m from S x S to S that satisfies the following conditions:
(1) S is a nonempty 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 nonempty subset L of S is a left ideal of S if
and only if SL is a subset of L. A nonempty subset R of S
is a right ideal of S if and only if RS is a subset of R.
A nonempty subset I of S is a (twosided) ideal of S if
and only if SI U IS is a subset of I. A compact semigroup
has minimal left, right, and twosided 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 onetoone 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 k1 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), BI1] 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
nonempty subset of S, then A is a semiideal of S if and
only if
(i) A is closed,
(ii) If x e S, then card[xA n
(SA)] e 1 and card[Ax n (SA)] 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 semiideal 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 semiideal
of S.
(c) If A is a semiideal 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 semiideal of G if
and only if card A = 1 or A = G.
(e) The intersection of a filter base of
semiideals of a compact semigroup is a semiideal of that
semigroup.
(f) If B is a subset of a compact semi
group, then there exists a semiideal minimal with respect
to containing B.
(g) Semiideals are preserved under
closed epimorphisms.
(h) Semiideals are not necessarily pre
served under inverses of continuous epimorphisms.
Proof: (a) If A is a semiideal of S and if x e S,
then x (SA()) r AA]. If x e (SAA()),
then xA c(SA) so that card xA = card[xA n(SA)] = 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 AA1l] 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 semiideal 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 semiideals
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 semiideal, 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 straightforward 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 semiideal 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 semiideal, but f (w)
is not a semiideal.
The following lemma is a restatement of Lemma 3.1.3.
3.1.9: Lemma. Let S and T be compact semigroups, let A be
a subsemigroup and semiideal 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 semiideal 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.
Semiideals 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 semiideal 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 semiideal 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 (SA).
According to (3.1), xy = xz so that the cardinality of
xA n (SA) 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 semiideal of A is a semiideal of S.
Proof: If P is a semiideal 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 semiideal 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 semiideal 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 AAl], 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 semiideal of S.
(b) If A is a closed subset of S and S
consists of right zeros, then A is semiideal 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
semiideal 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 semiideal
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 coordinatewise 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 coordinatewise 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 nonnegative 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 semiideal 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 semiideal 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 semiideal 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
coordinatewise 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 semiideal of K, then card A = 1
or A = K.
Proof: Let A be a semiideal 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 semiideal 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 semiideal 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 semiideal of I is a
semiideal 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 2simplicity. 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 nsimple if and only
if the cardinality of T is not less than n and every sub
set of T with n elements is a semiideal 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 2simple
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 2simple 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 2simple 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 2simple and x,y are two distinct
points of P. Since (x,y) is a semiideal, 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 semiideal of P. Thus,
P is 2simple.
3.4.4: Remark. A group is 2simple 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 2simple 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 semiideal of S and A satisfies conditions (1)
and (2), then every semiideal of A is a semiideal of S.
(1) If x e AA1, 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 2simple 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 semiideal 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 semiideal of A which is not a semiideal 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 semiideal of A because
card B = 2.
Suppose B is a semiideal 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 semiideal
of S.
The proof of (ii) follows in a similar
fashion.
3.4.7: Lemma. Let A be a 2simple 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 semiideal of A
which is not a semiideal of S.
(ii) If there exists a point x in A[ ]A
such that card Ax = 2, then there is a semiideal of A
which is not a semiideal 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 semiideal of A. If B is also a semiideal
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 semiideal of S.
(2) xa' = a' and xa" = a". If B = (a,a"), then B
is a semiideal since A is 2simple. Since xa" = a",
x e BB1). But xa = a' e B so that x BBI]. 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 2simplicity of A implies B
is a semiideal 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
semiideal of S.
The proof of (ii) is similar to the proof of (i).
3.4.8: Theorem. Let A be a 2simple subsemigroup of a
semigroup S such that the cardinality of A is larger than
2. Then every semiideal of A is a semiideal 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 semiideal 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 2simple 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 semiideal of A is a semiideal 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 2simple 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 semiideal of S.
3.4.11: Corollary. A semigroup T with cardinality great
er than two is 2simple 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 2simple,
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 semiideals 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 2simple, 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
2simple. 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 onetoone 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 nontrivial 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 coordinatewise 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 ml[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 nonempty
subsets of S. Then A is a Tideal if and only if TA U AT
is a subset of A, A is right Tideal if and only if AT is
a subset of A, and A is a left Tideal if and only if TA
is a subset of A. A Tideal 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 Tideal 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 Tideal is a retract of S. In section
4 it is shown that if L and R are minimal left and right
Tideals, respectively, and if both L and R are subsemi
groups, then the minimal Tideal LR is a retract of S in
case T is a homomorphic retract of S.
In Section 1 various examples of minimal Tideals
are presented and Section 2 contains some technical propo
sitions which are used in Sections 3 and 4.
1: Examples of Minimal Tideals.
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 Tideals 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 Qideal. If
T = x [l 0 then X x{ 0 is a minimal Tideal
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 Tideal
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 coordinatewise multiplication. If
T = X x (1) x Y, then the collection of minimal Tideals
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
Tideals 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 Tideal; if
Q = (1,1), then [3,w) is a minimal Qideal but neither T
nor is,}] 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 Tideal 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 Tideal 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 Tideals 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 coordinatewise 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 Tideal 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)
= f(1, [1 ], 1i, which is a contradiction.
2: Preliminary Propositions.
Throughout the rest of this chapter, S is a compact
semigroup, T is a nonempty 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 Tideal, 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 Tideal 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 Tideal, 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...
xnltnx = 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 Tideal, 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 Tideal, 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 Tideal 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 Tideal, then
f restricted to L is a onetoone 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 Tideal,
then f restricted to R is a onetoone function.
The proof is dual to the proof of Proposition 5.2.7.
5.2.10: Corollary. If L is a minimal left Tideal, if R
is a minimal right Tideal, and if L n R is not empty, then
f restricted to R is a onetoone 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 Tideal 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 onetoone,
(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 Tideal and if f
restricted to I is onetoone, 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 Tideal 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 onetoone 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 onetoone.
5.3.3: Theorem. If I is a minimal Tideal and if there
is a point a in I such that Ta is a subset of aT, then I
is a minimal left Tideal.
Proof: Since I is the union of minimal left Tideals [15],
there exists a minimal left Tideal 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 Tideal and if there
exists a e I such that aT is a subset of Ta, then I is min
imal right Tideal.
The proof is dual to that of Theorem 5.3.3.
5.3.5: Corollary. If I is a minimal Tideal 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 Tideal.
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 Tideal and hence by
5.2.8, f restricted to I is onetoone. Thus, 5.3.1 im
plies that I is a retract of S.
5.3.7: Corollary. If I is a minimal Tideal 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 Tideals and
the union of minimal right Tideals [17], there exist a
minimal left Tideal L and a minimal right Tideal 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 Tideal 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 Tideal 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 Iideal, 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 onetoone.
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 Tideal.
Therefore, f restricted to Tx onetoone implies that x = y.
4: The Subsemiqroup Case.
In this section, L and R are minimal left and right
Tideals, 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 Tideal. 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 Tideal 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 Tideal 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 Tideal 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
