Citation |

- Permanent Link:
- http://ufdc.ufl.edu/AA00032550/00001
## Material Information- Title:
- On lattice-ordered rings
- Creator:
- Neff, Mary Muskoff, 1930-
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 1956
- Language:
- English
- Physical Description:
- 27 leaves : ; 28 cm
## Subjects- Subjects / Keywords:
- Algebra ( jstor )
Bibliographies ( jstor ) Conceptual lattices ( jstor ) Equivalence relation ( jstor ) Graduates ( jstor ) Mathematical congruence ( jstor ) Mathematical sets ( jstor ) Mathematical theorems ( jstor ) Mathematics ( jstor ) Subrings ( jstor ) Lattice theory ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaf 26).
- General Note:
- Biography.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- |University of Florida
- Rights Management:
- This item is presumed in the public domain according to the terms of the Retrospective Dissertation Scanning (RDS) policy, which may be viewed at http://ufdc.ufl.edu/AA00007596/00001. The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
- Resource Identifier:
- 13038342 ( OCLC )
ocm13038342
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

ON LATTICE-ORDERED RINGS By MARY MUSKOFF NEFF A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1956 ,)s5 ACKNOWLEDGENTS The writer wishes to extend her thanks to the members of her supervisory committee, and particularly to Professor R. G. Blake, who directed the preparation of this dissertation. -ii - TABLE OF CONTENTS Page ACKNOWLEDGMENTS . . . . . . . . . . . . . . INTRODUCTION . . . . . . . . . . . . . . . Chapter I. PRELININARY CONCEPTS . . . . . . . II. LATTICE-ORDERED GROUPS . . . . . . III. LATTICE-ORDERED RINGS . . . . . IV. DISTRIBUTIVE LATTICE-ODERED RINGS BIBLIOGRAPHY . . . . . . . . .. .. . . BIOGIIAPTHICAL SKETCH . . . . . . . . . . . . * . * * * * . S * 0 0 0 3 ï¿½ 5 0 0 9 S. . . 14 . . . . 20 * 0 0 0 * 0 S S - iii - INTRODUCTION In modern mathematics an algebra usually means a set of elements together with a number, perhaps infinite, of single-valued finitary operations. A lattice, a type of partly ordered set, is usually classified as an algebra, as is a ring. A lattice-ordered ring, or 1-ring, is an algebra which is both a ring and a lattice, having then four operations-the two ring operations and the two lattice operations. In addition some specified relations involving a mixture of lattice and ring operations must hold. Rings have been studied extensively for many years, and for the last twenty years a steady flow of material concerning lattices has appeared. However the mathematical literature appears devoid of work on structures which are both lattices and rings, except for the case where the lattice is a chain. This work is concerned with some algebraic properties of 1-rings. Chapter I contains preliminary material chosen for its bearing on the later chapters. Definitions and theorems are given, with proofs generally omitted. Chapter II, which deals with lattice-ordered groups, is presented since a lattice-ordered ring is also a lattice-ordered group. Also many of the properties of -1- -2 lattice-ordered groups are used in obtaining the results on lattice-ordered rings. In Chapter III a lattice-ordered ring is defined following the manner of Garrett Birkhoff. The results of Chapter III and Chapter IV are believed to be new. CHAPTER I PRELIMINARY CONCEPTS Definition J . An algebra A is a set of elements together with an arbitrary number of operations f,. Each f. shall be a single-valued function assigning for a finite n = n(c), to every sequence (xl,'',xn) of n elements of A, a value f (xl,''',xn) in A. Definition 1.2. An equivalence relation is a binary relation - satisfying (1) For all x, x x. (Reflexive) (2) If x y, then y E x. (Symmetric) (3) If x y and y _ z, then x - z. (Transitive) Deflnition la. A congruence relation on an algebra A is an equivalence relation having the substitution property for each f., In other words, if xi = yi(o) (i = 1, , n) then f,(x,'',x) f (Y1,IYn)(e). Deflnition 1.4. A partly ordered set is a set with a binary relation < satisfying (1) For all x, x :S x. (Reflexive) (2) If x y and y :S x, then x = y. (Antisymmetric) (3) If x < y and y _S z, then x _ z. (Transitive) Definition .. If for every x and y in a partly ordered set either x 5 y or y _ x, then the set is a chain or a sigmly ordered set. -3- Definition 1.6. An upper bound to a subset X of a partly ordered set P is an element a in P such that x < a for every x in X. A least upper bound b is an upper bound such that b < a for every upper bound a. A lower bolud and a greatest lower bound may be defined dually. Definition 1.7. A lattice is a partly ordered set P, any two of whose elements have a greatest lower bound or meet, x A y, and a least upper bound or join, x V y. Theorem 1.1. In a lattice the two operations meet and join obey the following rules: (1) x A x x V x = x. (2) x A y y A x, and x V y = y V x. (3) x A (y A z) = (x A y) A z, and x V (y V z) - (x V y) V z. (4) x A (x V y) = x and x V (x A y) = x. Theorem 1.2. x < y and z < w imply x A z < y A w, and x V z< y V w. Deflnition 1.. A sublattice of a lattice is a subset which contains with any a, b, a. V b and a A b. Definition 1.L. A convex sublattice is a sublattice which contains with any a, b, all elements between a and b. Definition 1.. A complete lattice is a partly ordered set in which every set has a greatest lower bound and a least upper bound. Convention _j. If a lattice has a lower bound this element is denoted by 0. If a lattice has an upper bound this element is denoted by I. Definition L.1. A complement of an element x of a lattice L with 0 and I is an element y in L such that x A y = 0 and x V y = I; L is called comolemented if all its elements have complements. Definition . A lattice L is relatively c=plemented if given a < x _ b an element y exists in L such that x A y = a and x V y =b. Theorem d. Let A be any algebra, and let C be any set of congruence relations 3 on A. Define new relations , and r) by (1) a E b(g) means a 5 b(9) for all 6 in C. (2) a = b(T;) means that, for some finite sequence Xo, Xl, ***, x., a = x0, xm = b, and xi - () for some 0 in C. Then r is the greatest lower bound and ?) the least upper bound of the & in C, if C is partly ordered by letting l - 02 mean that the partition induced by &l is a subpartition of that induced by 92" Theorem 1.4. The congruence relations on any lattice L satisfy the infinite distributive laws A A -6- Theorem 1.,. Let L be the lattice of congruence relations on an algebra A. If B is obtained from A by introducing further finitary operations then the lattice of congruence relations on B is a sublattice of L. Definition 1,1j. A lattice is distributive if its elements satisfy x A (y V z) = (x A y) V (x A z). Theorem . A distributive lattice satisfies x V (y A z) = (x V y) A (x V z). Theorem J. A lattice is distributive if and only if x V y = x V z and x A y = x A z imply y = z. Definition 1,1. A Doolean algebra may be defined as a complemented distributive lattice. Definition a. By the set-theoretic union, X V Y, of two sets X and Y, is meant a set consisting of all the elements in either X or Y. The set-theoretic intersectjon, X n Y, is the set consisting of elements common to X and Y. Definition 2_). A non-empty set G of elements, together with an operation which assigns to every pair of elements a, b of G a unique value in G denoted a + b, is a grou- if the following postulates are fulfilled: (1) For all a, b, c, a + (b + c) = (a + b) + c. (2) There exists an element 0 in G such that 0 + a = a + 0 = a. (3) If a is in G there exists an element -a in G such that a + (-a) = -a + a = 0. -7- Definition J.A1Z A group is said to be commutative if a + b = b + a. Theorem 1.8. -(a + b) = -b - a. [-b - a is a short notation for -b + (-a).] DefinitIon a.. A subset S of a group G is a subgroup if it is itself a group with respect to the operation of G. Theorem 1.9. A subset S of a group G is a subgroup if and only if for any two elements a, b of S, a - b is in S. Definition J. If S is a subgroup of G and a is an element of G the set a + S (the set of elements a + s where s is in S) is a left coset. Similarly S + a is a right coset. A set that is both a left coset and a right coset is called a cq . Theorem 1.10. a + S and b + S are either identical or disjoint. Hence the various cosets of a group G with respect to a subgroup S constitute a partition of G. Definition J._2. A normal sub~roup N of a group G is a subgroup where a + N = N + a for every a in G. Theorem 111. The congruence relations on a group G are the partitions of G by the cosets of its different normal subgroups. Deflnition 2. A set of elements R, in which for any two elements a, b, a sum a + b and a product ab belonging to R is defined, is a ring provided (1) R is a commutative group with respect to the operation +. (2) a(bc) = (ab)c for all a. b, c in R. (3) a(b + c) = ab + ac and (b + c)a = ba + ca for alla, .b, c in R. Definition 1. A subset S of a ring R is a subring of R if S forms a ring under the operations of R. Theorem 1,12. A subset S of a ring R is a subring if and only if for a, b in S, a - b and ab are in S. Definition 1.23. A nonempty subset M of a ring R is an ideal if a and b in M imply a -b is in M and for a in M and r in R, ar and ra are in M. Theorem 1 . An ideal is a normal subgroup of R regarded as a commutative group. The congruence relations on a ring R are the partitions of R by the oosets of its different ideals. CHAPTER II IATTICE-ORDERED GROUPS Deflnition, .1. A set of elements G will be called a lattloe-ordered grouM, or an 1-group, if it satisfies the following postulates: (1) G is a group with respect to addition. (2) G is a latbice with respect to the operations V and A. (3) a + (b V c) = (a + b) V (a + c) and (b V c) + a = (b + a) V (c + a) for all a, b, c in G. Definition 2.2. An element of an 1-group will be called positive (negative) if a :t 0 (a < 0). Definition j_. The positive p. a of an element a is a V 0. The negative pj a- of a is a A 0. The absolule value lal of a is a V -a. The notation and nomenclature used here are largely due to Birkhoff [l].1 Theorem 2.1. a V b = -(-b A -a) and a A b --(-b V -a). Proof. Let x = a V b and y = -a A -b. Since x > a and x r b, -x < -a A -b = y. Further, y < -a and y <_.-b 1Numerals in square brackets refer to the bibliography concluding the dissertation. -9- - 10 - so that -y,2.a Vb=x. Thus x=-y, ora Vb = -(-b A -a). Dually a A b = -(-b V -a). Theorem2. a V b = a - (a A b) + b. Proof. a - (a A b) + b = [(a - a) V (a - b)] + b = b V a. Corollary 3. In a commutative 1-group a + b = (a V b) + (a A b). Corollary 24L2. a = a+ + a-. Proof. In Theorem 2.2 set b = 0 and rearrange terms. Theorem 2_. Any 1-group is a distributive lattice. Proof. To show that a lattice is distributive it is sufficient to establish that V b a V c and a A b = a A c imply b = c [i, p. 134]. From Theorem 2.2 b = (a A b) - a + (a V b) = (a A c) - a + (a V c) = c. Theorem 2,,4. If at least one of the elements a, b, c is positive then a V (b + c) < (a V b) + (a V c). Proof. (a V b) + (a V c) = [(a V b) + a] V [(a V b) + c] = (a + a) V (b + a) V (a + o) V (b + c). Now if a > 0, a + a > a and (a V b) + (a V c) >_. a V (b + c). Or if b > 0 then-b + a >_. a and the inequality holds. Similarly it holds if c > 0 for then a + c > a. Theorem Z.5. If a, b, and c are all positive elements then a A (b + c) Proof. (a A b) + (a A c) = (a + a) A (b + a) A (a + c) A (b + c). Since a, b, c are positive a + a > a, b + a > a, and a + c > a. Then (a A b) + (a A c) >a A a A a A (b + o) = a A (b + c). Theorem 2.6. If at least one of the elements a, b, c is negative then a A (b + c) > (a A b) + (a A c). Proof. (a A b) + (a A c) = (a + a) A (a + c) A (b + a) A (b + c). If a < 0 then a + a < a and (a A b) + (a A c) < a A (b + c). Similarly the inequality holds if either b or c is negative. Lemm . (-a)+ = -(a-). Proof. (-a)+ =-a V 0 = -(a A O) =-(a Theorem 2.7. al > 0. Proof. Obviously a V -a > a A -a. Then (a V -a) - (a A -a) = (a V -a) +(a V -a) >_ 0. But [(a V -a) A 0) + [(a V -a) A 0] = {[(a V -a) A 0) + (aV-a)) A {( a V -a) A 0. = [(a V -a) + (a V -a)] A (a V -a) A 0 = (a V -a) A 0. Subtraction of (a V -a) A 0 from both sides of the equation yields (a V -a) A 0 = 0. Hence Ial = a V -a > 0. Theorem Z,8. ja bi = (a V b) - (a A b). Proof. (a V b) - (a A b) = [(a V b) - a] V [(a V b) - b] = 0 V b - a V a - b V 0 = Ia - bl. Corollary aj. I aI = a+ - a-. Theorem 2.9. I(a V x) - (b V x) J _1a - bi. - 11 - - 12 - Proof. Expanding by Theorem 2.9, 1(a V x) - (b V x)J = (a V x V b V x) - [(a V x) A (b V x)] = ( a V b V x) - [( a A b) V (x A b) V (a A x) V x] = (a V b V x) - [(a A b) V x]. Similarly Ia - bl = (a V b) - (a A b). Now let a V b = y and a A b = y*, and let y - y* = t (note that t > 0). Then y V x (t + y*) V x = t + ly*V (-t+x)] < t + (y* V x). And(y V x)- (y* V x) < t. Returning to the original notation this yields (a V b V x)- [(a A b)V x] < (a V b) - (a A b), or I(a V x) - (b V x)I S Ia - bi. Theorem 2_q0. Ia V bi :_ Jal V Ibi; [a A bi < lal A Ibi. Proof. Ia V bl = a V b V -(a V b) = (a V b V -a) A (a V b V -b) < (a V b V -a V -b) A (a V b V -b V -a) = lal V Ibi. The second part of the theorem follows similarly. Theorem 2.11. In a commutative 1-group Ia + bl <_ lal + Ibi. Proof. Using Corollary 2.3 and Theorems 2.4 and 2.6, Ia + bl = (a + b)+ - (a + b)- < a+ + b+ - a- - b lal + Ibi. Definition 2.4. a and b are said to be orthogonal, in symbols a j b, provided lal A IbI = 0. Theorem 2.12. If a j b then lal + IbI = Ibi + lal. Proof. This follows immediately from Theorem 2.2 since b V a = a V b. - 13 Definitionafta. An 1-ideal of an 1-group G is a normal subgroup of G that contains with any a all x such that Ixi < lal. Theorem . The congruence relations on any 1-group are the partitions of G into the cosets of its different 1-ideals. (Proof omitted. See [1, p. 222]) CHAPTER III LA~rICE-ORDERED RINGS Definition 3.1. A set of elements R will be called a lattice-ordred rn,, or an 1-ring, if it satisfies the following postulates: (1) R is a ring with respect to addition and multiplication. (2) R is an 1-group uider addition. (3) The product of any two positive elements is positive. Since an 1-ring is also a commutative 1-group the results obtained for such 1-groups, some of which are outlined in the preceding chapter, will be valid for 1-rings. It is of interest to discuss relations involving the multiplication operation in addition to the lattice and group operations. The theorems of Chapter III refer to the elements of an 1-ring. Theorem 3.. The product of two negative elements is positive; the product of a positive and a negative element is negative. Proof. First, if a < 0 and b < 0 then -a > 0 and -b > 0. Thus ab = (-a)(-b) 0 . Second, if a < 0 and b > 0 then -a > 0 and -ab > 0 or ab < 0. Theorem ,. a > 0 and b > c imply ab 2 ac; a < 0 - 14 - - 15 - and b > c imply ab < ac. Proof. Since b > c, b - c > O. If a _ O, a(b - c) = ab - ac > 0, and ab > ac. If a < 0, a(b - c) = ab - ac < 0 and ab < ac. Theorem j7.. a+(b V c) > a+b V atc; a+(b A c) < a+b A a+c; a-(b V c) S a-b A a-c; a-(b A c)> a-b V a-c. Proof. a+(b V c) > a+b, and a+(b V c) a_ a+c. Hence a+(b V c) > a+b V a+c. Similar proofs may be written fathe remaining relations. Theorem 1.4. a-b+ + a+b- < ab < a+b+ + a-b-. Theorem 3a. labj <_ lalibi. Proof. labl = (ab)+ - (ab)- = (a+b+ + a-b- + a-b+ + a+b-)+ - (a+b+ + a-b- + a-b+ + a+b-)- < a+b+ + a-b- a-bt - ab- = (a+ - a-)(b+ - b-) = jallbl. Definition 3.2. If an 1-ring contains a unit, 1, aV a V 1 and aA = a A 1. Theorem 36. aV+ = a+V; aA+ - a+A; aA" = a-A = a-; av- O; a-V = 1; aA < bV; a- Proof. (a + b)V < aV + bV is a special case of Theorem 2.4. The equalities are a direct consequence of the fact that an 1-ring is a distributive lattice. Definition 3.. An element e of an 1-ring with a unit, 1, is called an ace if there exists an element e' in the 1-ring such that e V e' = 1 and e A e' = 0. Theorem 3._. If e is an ace and if el V e2 = e and eI A e2 = 0, then eI is an ace and eI' = e2 + e'. Proof. 0 < e2 A e' < e A el = 0, so that e2 + e' = (e2 V et) + (e2 A e') = e2 V e'. Then eI V eI' = el V (e2 V e') = e V e' = 1. In addition, 0 < el A ell = el A (e2 V e') = (el A e2) V (e, A e') = 0 V (e, A e') < e A e' = 0. Thus eI is an ace. Theorem .. The set of aces in an 1-ring form a Boolean algebra. Proof. If eI and e2 are aces then el V e2 is an ace, for (e, V e2)' = ell A e2'. Dually el A e2 is an ace. Theorem 3.10. For any ace e (i) e + e' = l; (ii) ee' = 0; and (iii) e2 = e. Proof. (W) e + e' = (e V e') + (e A e') = 1. (ii) ee' < e V ee' = el V ee' < e(l V e') = el = e. Similarly ee' < e'. So 0 < eel < e A e' = 0. (iii) e = e(e + e') = e2 + ee' = e2. Theorem 3,11. If e is an ace then (a) ex+ > (ex)+; (ii) ex- < (ex)-; (iii) exA < (ex )A; and (iv) ex x - ex. Proof. (f) ex+ = e(x V 0) > ex V 0 = (ex)+; (ii) ex- = e(x A 0) < ex A 0 = (ex)-; (iii) exA = e(x A 1) < ex A e < ex A 1 = (ex)A; and - 16 - - 17 - (iv) 0 < jexl A Ix - exi = jexj A letxl _s ejxj A e'Ix _S (e A e')jxJ = 0. Theorem 3.12. If el and e2 are aces and el A e2 - 0, then ele2 = 0. Proof. eI + e2 = el V e2 and thus is an ace. By Theorem 3.10 (iii), el + e2 = (eI + e2)2 = el2 + e22 + ele2 + e2eI = el + e2 + ele2 + e2e1. Hence eje2 + e2e1 0 0. Since e le2 0 and e2el 2_ 0, e1e2 = 0. Definition 3,.4. By an rl-ideal of an 1-ring R is meant a subset of R which is both an ideal of R regarded purely as a ring and an 1-ideal of R regarded as an 1-group. Definition 2 . A subset S of an 1-ring R is called an 1-subring if S is a subring of R viewed as a ring and a sublattice of R viewed as a lattice. Theorem 3_1. An rl-ideal M of an 1-ring R is a convex 1-subring. Proof. That M is a subring of R is evident since M is, by definition, an ideal of R. Now suppose a and b are in M. Since a V b = (a - b)+ + b, a V b will be in M provided (a - b)+ is in M. By Theorem 2.9 1(a - b)+l = l[(a - b) V 0] - CO V 031 :S I(a - b) - 01 = Ia - bl. Thus a V b is in M. Dually a A b is in M, and M is a sublattice of R. Suppose again that a and b are in M and b < x < a. Then IxI = x V -x :s a V -b < ja V -b and x is - 18 - in M. Hence M is a convex 1-subring. Theorem 3.14. The congruence relations on a 1-ring R are the partitions of R into the cosets of its rl-ideals. Proof. It was pointed out in Chapter I that the congruence relations on a ring are the partitions of the ring into the cosets of its different ideals. The congruence relations on an 1-ring R are those having the substitution property for the lattice operations of meet and join as well as for addition and multiplication. Recall the fact that a V b = (a - b)+ + b and that a A b = -(-a V -b). If an equivalence relation a has the substitution property for addition and multiplication then it will have it for joins and meets providing merely that a = b() implies a +- b(). Let M be an rl-ideal of R. If a - b is in M then, since la - b+l = I(a V 0) - (b V 0)1 :S ja - big a+- b+ is in N. In other words a S b mod M implies a b mod M. Conversely suppose M is the set of elements congruent to 0 under a congruence relation e. Then M is an ideal (ring) of R. And since M is evidently a normal subgroup of R then we have only to show that if a is in 14 and 1x1 :s lal then x is in M. jxj S lal implies that a A -a s x < a V -a. If x a b() then, since a = 0(e), 0 < b < 0. Hence x is in M, and M is an rl-ideal. This completes the proof. - 19 - Theorem 3.15. The rl-ideals of an 1-ring R form a complete distributive lattice, as do the congruence relations on R. Proof. It is obvious that the congruence relations on an 1-ring R are a subset of the congruence relations on R regarded purely as a lattice. That this subset is a complete sublattice is sho'm by Theorem 1.3, and this sublattice is distributive according to Theorem 1.4. CHAPTER IV DISTRIBUTIVE LATTICE-ORDERED RINGS Definition 1 A set of elements R will be called a distributive lattice-ordered Xn, or a dl-ring, if (1) R is an 1-ring. (2) a+(b V c) = a+b V ac and (b V c)a+ = ba+ V ca+ for all a, b, c in R. Consider the ring of n-dimensional vectors where a sum is defined in the usual fashion, the product of two vectors (al, a2, "', an) and (bl, b2, Q'', bn) is defined as (a 1b, a 2, , bn) and for i = 1, '" n the set of i-th components is a simply-ordered ring. If we latticeorder this ring by letting (a1, a2, ..., an) > 0 mean ai > 0 for each i, then the system forms a dl-ring. On the other hand the ring of n x n matrices A = [aij], ordered by letting A > 0 mean every aij 0, forms an 1-ring but not a dl-ring. The theorems of Chapter IV pertain to dl-rings and their elements. Theorem 4.1. a+(b A c) = a+b A a+c for all a, b, c in a dl-ring. Proof. a+(b A c) =-a+(-b V -c) = -(-a+b V a+c) = a+b A a+c. - 20 - - 21 - Theorem 42. () a-(b V c) = a-b A a-c; (ii) a-(b A c) = a-b V a-c. Proof. a-(b V c) = -(-a-)(b V c) = -(-a-b V -a-c) a-b A a-c, and thus (i) is proved. A proof for (ii) is similar. Definition 4.2. Two elements a and b are said to form an ordinary pair if they satisfy the inequality ab A ba < (a V b)(a A b) < ab V ba. Theorem 4.. If a and b are ordered they form an ordinary pair. Proof. Either a V b = a and a A b = b or a V b = b and a A b = a. Theorem 4.4. Every pair of positive elements (or of negative elements) forms an ordinary pair. Proof. If a and b are positive elements then (a V b)(a A b) = (a V b)a A (a V b)b > ba A ab. Also (a V b)(a A b) = a(a A b) V b(a A b) < ab V ba, and thus the theorem is established for positive elements. If a and b are negative elements (a V b)(a A b) = (a V b)a V (a V b)b = (aa A ba) V (ab A bb) < ba V ab. Also (a V b)(a A b) = a(a A b) A b(a A b) = (aa V ab) A (ba V bb) > ab A ba, and the proof is complete. Theorem 4.9. If a and b are orthogonal then ab and ba are orthogonal. Proof. According to Theorem 4.4, lal and IbI form - 22 - an ordinary pair. Using Theorem 3.5 we have 0 < labl A Ibal < lalIbi A Iblial : (jai V Ibi)(ial A ibi) = 0. Thus labi A ibal = 0. Theorem 46. If a and b are orthogonal then for every x and y in R, ax J by and xa J yb. Proof. Using Theorem 3.5, 0 -_ jax A ibyl < lalxi A IbilYjl :S al(Ixi V lyi) A lbl(Ixl V Iyj) = (lal A Ibi)(ixj V jyj) = 0. Hence ax I by. Similarly xaj yb. Corollary Ia. If a and b are members of a dl-ring with a unit and a and b are orthogonal, then for every x in R,a I bx and a i xb. Definition 43. A subset S of a dl-ring R is called a dl-subring if S is a dl-ring and if it is a sublattice of R viewed as a lattice. Definition 4.4 A subset of a dl-ring R with a unit that contains every element x of R such that x is orthogonal to the element a is denoted by *(a). Theorem 4.7. j(a) is a dl-subring. Proof. To show that j(a) is a subring it is sufficient to show that if x and y are elements of j(a) then x - y and xy are elements of i(a). Suppose x and y are in (a). Then, using Theorem 2.11 and Theorem 2.5, 0 < Ix - yj A lal :S (Ixi + lyl) A jal _5 (Ixl A lal) + (jyj A jal) = 0. Hence x - y is in i(a). And Corollary - 23 - 4.1 asserts that xy is in j(a). If x V y is also an element of j(a), then i(a) will be a sublattice. By Theorem 2.10, 0 < Ix V yl A lai S (lxi V Yl) A jal = (ixi A lal) V (jly A lai) = 0 V 0 = 0. Thus i(a) is a dl-subring. Theorem 4.8. 1(a) is a ring ideal. Proof. It follows directly from Corollary 4.1 that 1(a) is closed under both left and right multiplication by arbitrary elements of R. Theorem I_. i(a) is an rl-ideal. Proof. i(a) is obviously a normal subgroup. If x is an element of j(a) and y is an element of R such that IYI S jxi then 0 < lal A lYl S lal A Ixi = 0, and thus y is in '(a). Since by Theorem 4.8 j(a) is already a ring ideal, it is an rl-ideal. Theorem jO. If a > 0 and b > 0, 0 (a + b) i(a) n j(b). Proof. If x is in J(a + b) then 0 = lxi A Ja + bi lxi A lal 2 O. So x belongs to 1(a). Similarly x belongs to 1(b), and we have (a + b) Cj(a) rl1(b). Now suppose x is in both j(a) and 1(b). Then by Theorem 2.5 and Theorem 2.11, 0 < lxi A ja + bi S (lxi A lal) = (lxi A Ibl) = 0. Thus the theorem is established. Theorem 4.1. If a 2 0 and b > 0, J(a V b) = 1(a) t- 1(b). - 24 - Proof. Suppose x belongs to J(a V b). Then 0 = lxi A ja V bi = (lxj A a) V (lx A b). Since lxi A a > 0 and lxi A b > 0 we have that jx A a = 0 and lxi A b = 0. And J(a V b) C L(a) g j(b). On the other hand let x be in J(a) ) j(b). Then 0 < lxi A Ia V bi < Ix A al V Ix A bj = 0, and the proof is complete. Theorem 4j_ . For a > 0, laxi = ax, ax+ = (ax)+, and ax- = (ax)-. If a < 0, laxi = -aixi, ax+ = (ax)-, and ax- = (ax)+. Proof. Take a > 0. laxi ax V -ax = a(x V -x) = aixi. ax+ = a(x V 0) = ax V 0 = (ax)+. ax- = a(x A 0) = ax A 0 = (ax)-. Now take a :S 0. jaxi = ax V -ax - -a(-x V x) = -aixi. ax = a(x V 0) = ax A 0 = (ax)-. ax- = a(x A 0) = ax V 0 = (ax)+. Theorem jLU. If e is an ace, ex < eix S lxi, and exV < (ex)V. Proof. ex < lexi = elx S 1Xi = lxi. exV - e(x V 1) = ex V e < ex V 1 = (ex)V Theorem 4,14. Let X be a subset of R containing every x such that ax = xa for every a in R. X is a dl-subring. Proof. Suppose x and y are in X. Then (x - y)a - xa - ya = ax - ay = a(x - y) for every a in R. And xya = xay = axy for every a. Hence X is a subring. - 25 - Further (x V y)a = (x V y)(a+ + a-) = (xa+ V ya+) + (xa- A ya-) = (a+x V a+y) + (a-x A a-y) = a+(x V y) + a-(x V y) = a(x V y). Thus X is a dl-subring. Definition . In a dl-ring with a unit, where e is en ace, J(e) will denote a subset containing every element x of R such that x = ea = be for some a and b in R. Theory_ YZ. J(e) is a dl-subring. Proof. Suppose x =ea = be and y = ec =de are elements of J(e). Then x - y = ea - ec = e(a - c) and x - y = be - de = (b - d)e. Also xy = eade = e(ade) = (ead)e. So x - y and xy are in J(e) and J(e) is a subring. Also x V y = ea V ec = e(a V c) and x V y = be V de = (b V d)e. Thus x V y is in R. Theorem 4. If el and e2 are aces and e1 A e2 = 0 then J(el) (WJ(e2) = 1O1 (the set consisting only of the element 0). Proof. 0 is an element of J(el) since 0 = elO = Oe1. Similarly 0 is in J(e2). Suppose that at least one of the aces is 0, say e1 = 0. The theorem is trivially true since J(e1) = 0). Consider the case where eI 0 and e2 0. Suppose an element x # 0 is in J(el) n J(e2). Then x =ela = e2b for some a and b in R. Then, multiplying by el, e1e1a = e e 2b. By Theorem 2.10 e eI = e 1 and by Theorem 2.12 e1e2 = 0. So ela = Ob = 0. Hence x = e1a = 0, contrary to assumption. BIBLIOGRAPHY 1. Garrett Birkhoff, Lattice Theory, (revised edition), Am. Math. Soc., New [ork, 1949' 2. Garrett Birkhoff, Lattice-ordered Arous, Annals of Math. 43 (1942), pp. 299-331. 3. N. Bourbaki, Algbre, Actualites Scientifiques et Industrielles 1179, Paris, 1952. 4. Hans Hermes, Einffhrung in die Verbandstheorie, Berlin, 1955. 5. Hildegoro Nakano, Modem Sgectral Theory, Tokyo, 1950. 6. B. L. van der Waerden, Modern Algebra, Volume I, New York, 1949. - 26 - BIOGRAPHICAL SKETCH Mary Muskoff Neff was born January 20, 1930, in Jacksonville, Florida. She attended Purdue University from 1947 to 1952 and belonged to Delta Rho Kappa, scholastic honorary fraternity. She received a Bachelor of Science degree from Purdue in 1951. The following year she was a teaching assistant at Purdue, and in 1952 received the Master of Science degree there. After a year at Bell Telephone Laboratories, Murray Hill, New Jersey, she came to the University of Florida. She was a teaching assistant (1953 - 1954), a graduate fellow (1954 - 1955), and an instructor (1955 - 1956) in the Department of Mathematics. - 27 - This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of the committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 11, 1956. Dean, College of Arts and Sciences Dean, Graduate School SUPERVISORY COUNITTEE: Chairman Co-cha irmar Fm ,2 ,,__ -( |

Full Text |

- 17 -
(iv) O < |ex| A Â¡x ex| = jex| A |ex| < eÂ¡x[ A e'|x| < (e A e')|x) =0. Theorem 3.12. If e-^ and e2 are aces and e^ A e2 = 0, then e-j_e2 = * Proof. ei + e2 = el v e2 and thus is an ace. By Theorem 3*10 (iii), ei + e2 = (e-^ + e2)2 = e-^2 + e^2 + e-j_e2 + 2el = el + e2 + ele2 + e2el* Hence 6^2 + e2el = 0. Since e^e^ 0 and epeT > 0, ene 2 1 12 0. Definition 3.^. By an rl-ldeal of an 1-ring R is meant a subset of R which is both an ideal of R regarded purely as a ring and an 1-ideal of R regarded as an 1-group. Definition 3.3. A subset S of an 1-ring R is called an 1-subring if S is a subring of R viewed as a ring and a sublattice of R viewed as a lattice. Theorem 3.13. An rl-ldeal M of an 1-ring R is a convex 1-subring. Proof. That M is a subring of R is evident since M is, by definition, an ideal of R. Now suppose a and b are in M. Since a V b = (a b)+ + b, a V b will be in M provided (a b)+ is in M. By Theorem 2.9 |(a b)+| = I[(a b) V 0] [0 V 0]I < I(a b) o| = |a b|. Thus a V b is in M. Dually a A b is in M, and M is a sublattice of R. Suppose again that a and b are in M and b < x < a. Then |x| = x V -x < a V -b < |a V -b| and x is - 4- - Definition 1.6. An upper bound to a subset X of a partly ordered set P is an element a In P such that x < a for every x In X. A least upper bound b Is an upper bound such that b < a for every upper bound a. A lower bound and a greatest lower bound may be defined dually. ' Definition 1.7. A lattice is a partly ordered set P, any two of whose elements have a greatest lower bound or meet, x A y, and a least upper bound or join, x V y. Theorem 3,.l. In a lattice the two operations meet and join obey the following rules: (1) xAx=xVx=x. (2) x A y = y A x, and x V y = y V x. (3) x A (y A z) = (xA y) A z, and x V (y V z) = (x V y) V z. (4) x A (x V y) = x and x V (x A y) = x. Theorem 1*2. x < y and z < w imply x A z < y A w, and x V z < y V w. Definition l.g. A sublattice of a lattice Is a subset which contains with any a, b, a. V b and a A b. Definition 1.9. A convex sublattice is a sublat tice which contains with any a, b, all elements between a and b. Definition 1.10. A complete lattice is a partly ordered set in which every set has a greatest lower bound and a least upper bound. CHAPTER II LATTICE-ORDERED GROUPS Definition 2.1. A set of elements G will be called a lattloe-ordered group, or an 1-group, if it sat isfies the following postulates: (1) G is a group with respect to addition. (2) G is a lattice with respect to the operations V and A. (3) a + (b V c) = (a + b) V (a + c) and (b V c) + a = (b + a) V (c + a) for all a, b, c in G. Definition 2.2. An element of an 1-group will be called positive (negative) if a > 0 (a < 0). Definition 2.3 The positive part a+ of an element a is a V 0. The negative part a of a is a A 0. The absolute value |a| of a is a V -a. The notation and nomenclature used here are largely due to Birkhoff [I].-* Theorem 2.1. a V b = -(-b A -a) and a A b = -(-b V -a). Proof. Let x = a V b and y = -a A -b. Since x > a and x 2T b, -x < -a A -b = y. Further, y < -a and y < -b Numerals in square brackets refer to the bibliog raphy concluding the dissertation. - 9 - - 15 - and b > c imply ab < ac. Proof. Since b>c, b-c>0. If a > 0, a(b c) = ab ac > 0, and ab > ac. If a < 0, a(b c) = ab ac < 0 and ab < ac. Theorem 1.1. a+(b V c) > a+b V a+c; a+(b A c) < a+b A a+c; a(b V c) < a~b A a"c; a~(b A c) > a"b V ao. Proof. a+(b V c) > a+b, and a+(b V c) > a+c. Hence a+(b V c) > a+b V a+c. Similar proofs may be written fcrthe remaining relations. Theorem 1.4. a""b+ + a+b < ab < a+b+ + a**b*. Theorem Iab| < |ai ibl. Proof, jab| = (ab)+ (ab)" = (a+b+ + a~b~ + ab+ + a+b~)+ (a+b+ + ab~ + ab+ + a+b) < a+b+ + a"*b - a"b+ a+b" = (a+ a**) (b+ b) = |a| jbj . Definition 1.2. If an 1-rlng contains a unit, 1, aV = a V 1 and aA = a A 1. Theorem 1^6. av+ = a+v; aA+ = a+A; aA = aA a"; av* = 0; a= 1; aA < bv; a < bv; a < aA; a+ < av. Theorem 1.7. (a + b)v < av + bv; (a V b)A = aA V bA; ( a A b)v = av A bv. Proof, (a + b)v < av + bv is a special case of Theorem 2,4. The equalities are a direct consequence of the fact that an 1-ring is a distributive lattice. Definition^ 1.1. An element e of an 1-ring with a unit, 1, is called an ace if there exists an element e' in 10 so that -y > a V b = x. Thus x = -y, or a V b = -(-b A -a). Dually a A b = -(-b V -a). Theorem 2.2. a V b = a (a A b) + b. Proof, a (a A b) + b = [(a a) V (a b)] + b = b V a. nvh}' > Corollary 2^1. In a commutative 1-group a + b = (a V b) + (a A b). Corollary 2.2. a = a+ + a. Proof. In Theorem 2.2 set b = 0 and rearrange terms. Theorem 2.3. Any 1-group is a distributive lattice. Proof. To show that a lattice is distributive it is sufficient to establish that .VJb = a V c and a A b Â£ Th L7 jP-- = a A c imply b = c [1, p. 13^3. From Theorem 2.2 b = (a A b) a + (a V b) = (a A c) a + (a V c) = c. Theorem 2.4. If at least one of the elements a, b, c is positive then a V (b + c) < (a V b) + (a V c). Proof, (a V b) + (a V c) = [(a V b) + a] V [(a V b) + c] = (a + a) V (b + a) V (a + c) V (b + c). Now if a > 0, a+a>a and (a V b) + (a V c) > a V (b + c). Or if b > 0 then b + a > a and the inequality holds. Similarly it holds if c > 0 for then a + c > a. Theorem 2.5. If a, b, and c are all positive elements then a A (b + c) < (a A b) + (a A c). BIOGRAPHICAL SKETCH Mary Muskoff Neff was bom January 20, 193> in Jacksonville, Florida. She attended Purdue University from 19^7 to 1952 and belonged to Delta Bho Kappa, scho lastic honorary fraternity. She received a Bachelor of Science degree from Purdue in 1951. The following year she was a teaching assistant at Purdue, and in 1952 re ceived the Master of Science degree there. After a year at Bell Telephone Laboratories, Murray Hill, New Jersey, she came to the University of Florida. She was a teaching assistant (1953 195*0 > a graduate fellow (195^ 1955) and an instructor (1955 - 1956) in the Department of Mathematics. - 27 - - 5 - Convention 1.3,. If a lattice has a lower bound this element is denoted by 0. If a lattice has an upper bound this element is denoted by I. Definition 1.11. A complement of an element x of a lattice L with 0 and I is an element y in L such that x A y = 0 and x V y = I; Lis called complemented if all its elements have complements. Definition 1.12. A lattice L is relatively com plemented if given a < x < b an element y exists in L such that x A y = a and x V y = b. Theorem l.~5. Let A be any algebra, and let C be any set of congruence relations S on A. Define new rela tions j and 09 by (1) a 2 b(j) means a = b(6) for all 0 in C. (2) a = b(r;) means that, for some finite sequence x0- xl> '"> xm- a = x0- = b' ld xi-l = xi^l> fOT some 9^ in C, Then f is the greatest lower bound and 7) the least upper bound of the Â£ in C, ifCis partly ordered by letting 6^ < mean that the partition induced by 0^ is a sub partition of that induced by S^. Theorem 1.4. The congruence relations on any lattice L satisfy the infinite distributive laws e r\ V6>a = v(<9 /I 0a). A A ON LATTICE-ORDERED RINGS By MARY MUSKOFF NEFF A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1956 ACKNOWLEDGMENTS 'The x^rlter wishes to extend her thanks to the members of her supervisory committee, and particularly to Professor R. G. Blake, who directed the preparation of this dissertation. - ii - TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii INTRODUCTION 1 Chapter I. PRELIMINARY CONCEPTS 3 II. LATTICE-ORDERED GROUPS 9 III. LATTICE-ORDERED RINGS l4- IV. DISTRIBUTIVE LATTICE-ORDERED RINGS .... 20 BIBLIOGRAPHY 26 BIOGRAPHICAL SKETCH 27 \ ill INTRODUCTION In modern mathematics an algebra usually means a set of elements together with a number, perhaps infinite, of single-valued finitary operations. A lattice, a type of partly ordered set, is usually classified as an algebra, as is a ring. A lattice-ordered ring, or 1-ring, is an algebra which is both a ring and a lattice, having then four oper ationsthe two ring operations and the two lattice oper ations. In addition some specified relations involving a mixture of lattice and ring operations must hold. Rings have been studied extensively for many years, and for the last twenty years a steady flow of material concerning lattices has appeared. However the mathemati cal literature appears devoid of work on structures which are both lattices and rings, except for the case where the lattice is a chain. This work is concerned with some algebraic prop erties of 1-rings, Chapter I contains preliminary mate rial chosen for its bearing on the later chapters. Def initions and theorems are given, with proofs generally omitted. Chapter II, which deals with lattice-ordered groups, is presented since a lattice-ordered ring is also a lattice-ordered group. Also many of the properties of 1 2 lattice-ordered groups are used in obtaining the results on lattice-ordered rings. In Chapter III a lattice-ordered ring is defined following the manner of Garrett Birkhoff. The results of Chapter III and Chapter IV are believed to be new. CHAPTER I PRELIMINARY CONCEPTS Definition 1.1. An algebra A Is a set of elements together with an arbitrary number of operations fa. Each fa shall be a single-valued function assigning for a finite n = n(a), to every sequence (x^,***,xn) of n ele ments of A, a value f^ix^,* *,x^) in A. Definition 1.2. An equivalence relation is a binary relation = satisfying (1) For all x, x = x. (Reflexive) (2) If x = y, then y = x. (Symmetric) (3) If x = y and y = z, then x = z. (Transitive) Definition 1.1. A congruence relation on an algebra A Is an equivalence relation having the substitu tion property for each fa. In other words, if x^ = y^isO (i = 1, , n) then f^x^ ,xn) = fa(y1, * ,yn) (). Definition 1.4. A partly ordered set is a set with a binary relation < satisfying (1) For all x, x < x. (Reflexive) (2) If x < y and y < x, then x = y. (Antisymmetric) (3) If x < y and y < z, then x < z. (Transitive) Definition 1.5. If for every x and y in a partly ordered set either x < y or y < x, then the set is a chain or a simply ordered set. - 3 - - 4- - Definition 1.6. An upper bound to a subset X of a partly ordered set P is an element a In P such that x < a for every x In X. A least upper bound b Is an upper bound such that b < a for every upper bound a. A lower bound and a greatest lower bound may be defined dually. ' Definition 1.7. A lattice is a partly ordered set P, any two of whose elements have a greatest lower bound or meet, x A y, and a least upper bound or join, x V y. Theorem 3,.l. In a lattice the two operations meet and join obey the following rules: (1) xAx=xVx=x. (2) x A y = y A x, and x V y = y V x. (3) x A (y A z) = (xA y) A z, and x V (y V z) = (x V y) V z. (4) x A (x V y) = x and x V (x A y) = x. Theorem 1*2. x < y and z < w imply x A z < y A w, and x V z < y V w. Definition l.g. A sublattice of a lattice Is a subset which contains with any a, b, a. V b and a A b. Definition 1.9. A convex sublattice is a sublat tice which contains with any a, b, all elements between a and b. Definition 1.10. A complete lattice is a partly ordered set in which every set has a greatest lower bound and a least upper bound. - 5 - Convention 1.3,. If a lattice has a lower bound this element is denoted by 0. If a lattice has an upper bound this element is denoted by I. Definition 1.11. A complement of an element x of a lattice L with 0 and I is an element y in L such that x A y = 0 and x V y = I; Lis called complemented if all its elements have complements. Definition 1.12. A lattice L is relatively com plemented if given a < x < b an element y exists in L such that x A y = a and x V y = b. Theorem l.~5. Let A be any algebra, and let C be any set of congruence relations S on A. Define new rela tions j and 09 by (1) a 2 b(j) means a = b(6) for all 0 in C. (2) a = b(r;) means that, for some finite sequence x0- xl> '"> xm- a = x0- = b' ld xi-l = xi^l> fOT some 9^ in C, Then f is the greatest lower bound and 7) the least upper bound of the Â£ in C, ifCis partly ordered by letting 6^ < mean that the partition induced by 0^ is a sub partition of that induced by S^. Theorem 1.4. The congruence relations on any lattice L satisfy the infinite distributive laws e r\ V6>a = v(<9 /I 0a). A A - 6 - Theorem 1.5. Let L be the lattice of congruence relations on an algebra A. If B is obtained from A by- introducing further finitary operations then the lattice of congruence relations on B is a sublattice of L. Definition 1.15. A lattice is distributive if its elements satisfy x A (y V z) = (x A y) V (x A z). Theorem 1.6. A distributive lattice satisfies x V (y A z) = (x V y) A (x V z). Theorem 1.7. A lattice is distributive if and only ifxVy=xVz and x A y = x A z imply y = z. Definition l.l4. A Boolean algebra may be defined as a complemented distributive lattice. Definition 1.15. By the set-theoretic union. XU Y, of two sets X and Y, is meant a set consisting of all the elements in either X or Y. The set-theoretic Intersection,. X A Y, is the set consisting of elements common to X and Y, Definition l.lo. A non-empty set G of elements, together with an operation which assigns to every pair of elements a, b of G a unique value in G denoted a + b, is a group if the following postulates are fulfilled: (1) For all a, b, c, a + (b + c) = (a + b) + c. (2) There exists an element 0 in G such that 0 + a = a + 0 = a. (3) If a is in G there exists an element -a in G such that a + (-a) = -a + a = 0 - 7 - Definition 1.17. A group is said to be commutative if a + b = b + a. Theorem l.g. -(a + b) = -b a. [-b a is a short notation for -b + (-a).] Definition l.lg. A subset S of a group G is a subgroup if it is itself a group with respect to the oper ation of G. Theorem 1*1- A subset S of a group G is a sub group if and only if for any two elements a, b of S, a b is in S. Definition 1.19. If S is a subgroup of G and a is an element of G the set a + S (the set of elements a + s where s is in S) is a left coset. Similarly S + a is a right coset. A set that is both a left coset and a right coset is called a coset. Theorem 1.10. a + S and b + S are either identical or disjoint. Hence the various cosets of a group G with respect to a subgroup S constitute a partition of G. Definition 1.20. A normal subgroup N of a group G is a subgroup where a + N N + a for every a in G. Theorem 1.11. The congruence relations on a group G are the partitions of G by the cosets of its different normal subgroups. Definition 1.21. A set of elements R, in which for any two elements a, b, a sum a + b and a product ab belonging to R is defined, is a ring provided (1) R is a commutative group with respect to the operation + . (2) a(bc) = (ab)c for all a, b, c in R. (3) a(b + c) = ab + ac and (b + c)a = ba + ca for all a, b, c in R. Definition 1.22. A subset S of a ring R is a subring of R if S forms a ring under the operations of R. Theorem 1.12. A subset S of a ring R is a subring if and only if for a, b in S, a b and ab are in S. Definition 1.23. A nonempty subset M of a ring R is an ideal if a and b in M imply a b is in M and for a in M and r in R, ar and ra are in M, Theorem 1.13. An ideal is a normal subgroup of R regarded as a commutative group. The congruence relations on a ring R are the partitions of R by the oosets of its different ideals. CHAPTER II LATTICE-ORDERED GROUPS Definition 2.1. A set of elements G will be called a lattloe-ordered group, or an 1-group, if it sat isfies the following postulates: (1) G is a group with respect to addition. (2) G is a lattice with respect to the operations V and A. (3) a + (b V c) = (a + b) V (a + c) and (b V c) + a = (b + a) V (c + a) for all a, b, c in G. Definition 2.2. An element of an 1-group will be called positive (negative) if a > 0 (a < 0). Definition 2.3 The positive part a+ of an element a is a V 0. The negative part a of a is a A 0. The absolute value |a| of a is a V -a. The notation and nomenclature used here are largely due to Birkhoff [I].-* Theorem 2.1. a V b = -(-b A -a) and a A b = -(-b V -a). Proof. Let x = a V b and y = -a A -b. Since x > a and x 2T b, -x < -a A -b = y. Further, y < -a and y < -b Numerals in square brackets refer to the bibliog raphy concluding the dissertation. - 9 - 10 so that -y > a V b = x. Thus x = -y, or a V b = -(-b A -a). Dually a A b = -(-b V -a). Theorem 2.2. a V b = a (a A b) + b. Proof, a (a A b) + b = [(a a) V (a b)] + b = b V a. nvh}' > Corollary 2^1. In a commutative 1-group a + b = (a V b) + (a A b). Corollary 2.2. a = a+ + a. Proof. In Theorem 2.2 set b = 0 and rearrange terms. Theorem 2.3. Any 1-group is a distributive lattice. Proof. To show that a lattice is distributive it is sufficient to establish that .VJb = a V c and a A b Â£ Th L7 jP-- = a A c imply b = c [1, p. 13^3. From Theorem 2.2 b = (a A b) a + (a V b) = (a A c) a + (a V c) = c. Theorem 2.4. If at least one of the elements a, b, c is positive then a V (b + c) < (a V b) + (a V c). Proof, (a V b) + (a V c) = [(a V b) + a] V [(a V b) + c] = (a + a) V (b + a) V (a + c) V (b + c). Now if a > 0, a+a>a and (a V b) + (a V c) > a V (b + c). Or if b > 0 then b + a > a and the inequality holds. Similarly it holds if c > 0 for then a + c > a. Theorem 2.5. If a, b, and c are all positive elements then a A (b + c) < (a A b) + (a A c). - 11 - Proof, (a A b) + (a A c) = (a + a) A (b + a) A (a + c) A (b + c). Since a, b, c are positive a + a > a, b + a > a, and a + c > a. Then (a A b) + (a A c) > a A a A a A (b+c) = a A (b+c). Theorem 2.6. If at least one of the elements a, b, c is negative then a A (b + c) > (a A b) + (a A c). Proof, (a A b) + (a A c) = (a + a) A (a + c) A (b + a) A (b + c). If a < 0 then a + a < a and (a A b) + (a A c) < a A (b+c). Similarly the inequality holds if either b or c is negative. Lemma 2.1. (-a)+ = -(a"). Proof, (-a)+ = -a V 0 = -(a A 0) = -(a-). Theorem 2*1. |a| > 0. Proof. Obviously a V -a > a A -a. Then (a V -a) (a A -a) = (a V -a) + (a V -a) > 0. But [(a V -a) A 0] + [(a V -a) A 0] = {[(a V -a) A 0] + (aV -a)} A {( a V -a) A 0} = [(a V -a) + (a V -a)] A (a V -a) A 0 = (a V -a) AO. Subtraction of (a V -a) A 0 from both sides of the equation yields (a V -a) A 0 = 0. Hence I a| = a V -a > 0. Theorem 2.S. |a b| = (a V b) (a A b). Proof, (a V b) (a A b) = [(a V b) a] V [(a V b) b] = 0 V b a V a b V 0 = Â¡a b|. Corollary 2.3. |a| = a+ a". Theorem 2.9. |(a V x) (b V x)| < |a b|. - 12 Proof. Expanding by Theorem 2.S, |(a V x) (b V x)| =(aVxVbVx)- C(a V x) A (b V x)] = ( a V b V x) - [( a A b) V (x A b) V (a A x) V x] = (a V b V x) - [(a A b) V x]. Similarly Â¡a b| = (a V b) (a A b). Now let a V b = y and a A b = y*, and let y y* = t (note that t > 0). Then y V x = (t + y*) V x = t + [y* v (-t + x) ] < t + (y* V x). And (y V x} (y* V x) < t. Returning to the original notation this yields (a V b V x)- [(a A b)V x] < (a V b) (a A b), or |(a V x) (bVx)| < |a b|. Theorem 2.10. |aVb| <|aÂ¡ V |b|; |aAb| * < |aj A Â¡b1. Proof. |aVb| =aVbV -(a V b) = (a V b V -a) A (a V b V -b) < (a V b V -a V -b) A (a V b V -b V -a) = |a| V |bi. The second part of the theorem follows similarly. Theorem 2.11. In a commutative 1-group I a + b | < | a | + | b ( . Proof. Using Corollary 2.3 and Theorems 2.4 and 2.6, |a + bÂ¡ = (a + b)+ (a + b)~ < a+ + b+ a" b = I a. | + | b | . Definition 2.4. a and b are said to be orthogonal, in symbols a J_ b, provided |a| A |b| =0. Theorem 2.12. If a j. b then |a| + |b| = |b| + Â¡a| . Proof. This follows immediately from Theorem 2.2 since b V a = a V b. - 13 - Definition 2.5. An 1-ldeal of an 1-group G is a normal subgroup of G that contains with any a all x such that |x| < |a|. Theorem 2.15. The congruence relations on any 1-group are the partitions of G into the cosets of its different 1-ideals. (Proof omitted. See [1, p. 222]) CHAPTER III LATTICE-ORDERED RINGS Definition 3.1. A set of elements R will be called a lattice-ordered ring, or an l-rlngf if It satisfies the following postulates: (1) R is a ring with respect to addition and multipli cation. (2) R is an 1-group under addition. (3) The product of any two positive elements is positive. Since an 1-ring is also a commutative 1-group the results obtained for such 1-groups, some of which are out lined in the preceding chapter, will be valid for 1-rings. It is of interest to discuss relations involving the multi plication operation in addition to the lattice and group operations. The theorems of Chapter III refer to the elements of an 1-ring. Theorem 1*1. The product of two negative elements is positive; the product of a positive and a negative ele ment is negative. Proof. First, if a < 0 and b < 0 then -a > 0 and -b > 0. Thus ab = (-a)(-b) >0. Second, if a < 0 and b > 0 then -a > 0 and -ab >0 or ab < 0. Theorem 3.2. a > 0 and b > c imply ab > ac; a < 0 - 14 .- - 15 - and b > c imply ab < ac. Proof. Since b>c, b-c>0. If a > 0, a(b c) = ab ac > 0, and ab > ac. If a < 0, a(b c) = ab ac < 0 and ab < ac. Theorem 1.1. a+(b V c) > a+b V a+c; a+(b A c) < a+b A a+c; a(b V c) < a~b A a"c; a~(b A c) > a"b V ao. Proof. a+(b V c) > a+b, and a+(b V c) > a+c. Hence a+(b V c) > a+b V a+c. Similar proofs may be written fcrthe remaining relations. Theorem 1.4. a""b+ + a+b < ab < a+b+ + a**b*. Theorem Iab| < |ai ibl. Proof, jab| = (ab)+ (ab)" = (a+b+ + a~b~ + ab+ + a+b~)+ (a+b+ + ab~ + ab+ + a+b) < a+b+ + a"*b - a"b+ a+b" = (a+ a**) (b+ b) = |a| jbj . Definition 1.2. If an 1-rlng contains a unit, 1, aV = a V 1 and aA = a A 1. Theorem 1^6. av+ = a+v; aA+ = a+A; aA = aA a"; av* = 0; a= 1; aA < bv; a < bv; a < aA; a+ < av. Theorem 1.7. (a + b)v < av + bv; (a V b)A = aA V bA; ( a A b)v = av A bv. Proof, (a + b)v < av + bv is a special case of Theorem 2,4. The equalities are a direct consequence of the fact that an 1-ring is a distributive lattice. Definition^ 1.1. An element e of an 1-ring with a unit, 1, is called an ace if there exists an element e' in -li the 1-ring such that e V e* = 1 and e A e =0. Theorem 3.g. If e is an ace and if e-^ V ej = e and e-j. A e2 = 0, then e-j. is an ace and e-^' =62 + 6*. Proof. 0 < e2 A e1 < e A e* =0, so that e2 + e' = (e2 V e*) + (e2 Ae1) = e2 V e*. Then e^ V e-^' = e^ V (e2 V e') = e V e' =1. In addition, 0 < e-^ A e^' = A (e2 V e') = (e^ A e2) V (e-^ A e') = 0 V (e^ A e1) < e a e' =0. Thus e1 is an ace. Theorem 3.9. The set of aces in an 1-ring form a Boolean algebra. Proof. If e^ and e2 are aces then e2 V e2 is an ace, for (e^ V e2)' = e^ A e2'. Dually e^ A e2 is an ace. Theorem 3.10. For any ace e(i)e+e'=l; (ii)ee' =0; and (iii) e2 = e. Proof. (i) e+ e' = (e V e') + (e A e') =1. (ii) ee' < e V ee* = el V ee' < e(l V e') = el = e. Similarly ee1 < e*. So 0 < ee' < e A e' = 0, (iii) e = e(e + e') = e2 + ee' = e2. Theorem 3.11. If e is an ace then (i) ex+ > (ex)+; (ii) ex" < (ex)"; (iii) ex^ < (ex)A; and (iv) ex J_ x ex. Proof, (i) ex+ = e(x V 0) > ex V 0 = (ex)+; (ii) ex = e(x A0) < ex A 0 = (ex); (iii) ex^ =e(x A 1) < ex A e < ex A 1= (ex)A; and - 17 - (iv) O < |ex| A Â¡x ex| = jex| A |ex| < eÂ¡x[ A e'|x| < (e A e')|x) =0. Theorem 3.12. If e-^ and e2 are aces and e^ A e2 = 0, then e-j_e2 = * Proof. ei + e2 = el v e2 and thus is an ace. By Theorem 3*10 (iii), ei + e2 = (e-^ + e2)2 = e-^2 + e^2 + e-j_e2 + 2el = el + e2 + ele2 + e2el* Hence 6^2 + e2el = 0. Since e^e^ 0 and epeT > 0, ene 2 1 12 0. Definition 3.^. By an rl-ldeal of an 1-ring R is meant a subset of R which is both an ideal of R regarded purely as a ring and an 1-ideal of R regarded as an 1-group. Definition 3.3. A subset S of an 1-ring R is called an 1-subring if S is a subring of R viewed as a ring and a sublattice of R viewed as a lattice. Theorem 3.13. An rl-ldeal M of an 1-ring R is a convex 1-subring. Proof. That M is a subring of R is evident since M is, by definition, an ideal of R. Now suppose a and b are in M. Since a V b = (a b)+ + b, a V b will be in M provided (a b)+ is in M. By Theorem 2.9 |(a b)+| = I[(a b) V 0] [0 V 0]I < I(a b) o| = |a b|. Thus a V b is in M. Dually a A b is in M, and M is a sublattice of R. Suppose again that a and b are in M and b < x < a. Then |x| = x V -x < a V -b < |a V -b| and x is - IS - in M. Hence M is a convex 1-subring. Theorem 1 ,l4. The congruence relations on an 1-ring R are the partitions of R into the cosets of its rl-ideals. Proof. It was pointed out in Chapter I that the congruence relations on a ring are the partitions of the ring into the cosets of its different ideals. The congru ence relations on an 1-ring R are those having the substi tution property for the lattice operations of meet and join as well as for addition and multiplication. Recall the fact that a V b = (a b)+ + b and that a A b = -(-a V -b). If an equivalence relation Q has the substitution property for addition and multiplication then it will have it for joins and meets providing merely that a = b(<9) implies a+ = b+(0). Let M be an rl-ideal of R. If a b is in M then, since |a+ b+| = |(a V 0) (b V 0)| < Â¡a-b|, a+ b+ is in M. In other words a = b mod M implies a+ = mod M. Conversely suppose M is the set of elements congruent to 0 under a congruence relation O. Then M is an ideal (ring) of R. And since M is evidently a normal subgroup of R then we have only to show that if a is in M and W < I a I then x is in M. |x| < |a| implies that aA-a completes the proof. - 19 - Theorem 3.15. The rl-ideals of an 1-ring R form a complete distributive lattice, as do the congruence relations on R. Proof. It is obvious that the congruence relations on an 1-ring R are a subset of the congruence relations on R regarded purely as a lattice. That this subset is a complete sublattice is shown by Theorem 1.3, and this sublattice is distributive according to Theorem 1.4, CHAPTER IV DISTRIBUTIVE LATTICE-ORDERED RINGS Definition 4.1. A set of elements R will be called a distributive lattice-ordered ringr or a dl-rlngr if (1) R is an 1-ring. (2) a+(b V c) = a+b V a+c and (b V c)a+ = ba+ V ca+ for all a, b, c In R. Consider the ring of n-dimensional vectors where a sum is defined in the usual fashion, the product of two vectors (a^ a2, ***, an) and (b^, b2> bn) is defined as (a1b1, a2b2, ***, a^t^) and for i = 1, n the set of i-th components is a simply-ordered ring. If we lattice- order this ring by letting (a^, a2, **, a^) > 0 mean aj_ > 0 for each i, then the system forms a dl-rlng. On the other hand the ring of n x n matrices A = [a^], ordered by letting A > 0 mean every a >0, forms an 1-ring but not J. J a dl-ring. The theorems of Chapter IV pertain to dl-rings and their elements. Theorem 4.1. a+(b A c) = a+b A a+c for all a, b, c In a dl-ring. Proof. a+(b A c) =-a+(-b V -c) = -(-a+b V a+c) = a+b A a+c. 20 - 21 Theorem 4.2. (i) a(b V c) = ab A ac; (11) a(b A c) = ab V ac. Proof. a~(b V c) = -(-a~)(b V c) = -(-ab V -ac) = ab A ac, and thus (i) is proved. A proof for (li) Is similar. Definition 4.2. Two elements a and b are said to form an ordinary pair if they satisfy the inequality ab A ba < (a V b)(a A b) < ab V ba. Theorem 4.1. If a and b are ordered they form an ordinary pair. Proof. Either a V b = a and a A b = b or a V b = b and a A b = a. Theorem 4.4. Every pair of positive elements (or of negative elements) forms an ordinary pair. Proof. If a and b are positive elements then (a V b)(a A b) = (a V b)a A (a V b)b > ba A ab. Also (a V b)(a A b) = a(a A b) V b(a A b) < ab V ba, and thus the theorem is established for positive elements. If a and b are negative elements (a V b)(a A b) = (a V b)a V (aVb)b=(aaA ba) V (ab A bb) < ba V ab. Also (a V b)(a A b) = a(a A b) A b(a A b) = (aa V ab) A (ba V bb) > ab A ba, and the proof is complete. Theorem If a and b are orthogonal then ab and ba are orthogonal. Proof. According to Theorem 4.4, |a| and |b| form - 22 an ordinary pair. Using Theorem 3*5 we have 0 < |ab| A Â¡ba| < 1 aj |b1 A |b)|a| < ( | a | V |b[)(Â¡a| A |b|) =0. Thus |ab| A Â¡ba| = 0. Theorem 4.6. If a and b are orthogonal then for every x and y in R, ax 1 by and xa 1 yb. Proof. Using Theorem 3*5 0 < Â¡ax| A j by | < I a||x| A j b| |y| < ia|(|xi V |yÂ¡) A |b|(|x| V |y|) = ( J a! A Â¡b j ) ( | x| V | y | ) =0. Hence ax J_ by. Similarly xa 1 yb. Corollary iui. If a and b are members of a dl-ring with a unit and a and b are orthogonal, then for every x in R,a 1 bx and a J. xb. Definition 4.3. A subset S of a dl-ring R is called a dl-subrlng if S is a dl-ring and if it is a sublattice of R viewed as a lattice. Definition 4,4. A subset of a dl-ring R with a unit that contains every element x of R such that x is orthogonal to the element a is denoted by 1(a) Theorem 4.7. 1(a) is a dl-subring. Proof. To show that 1(a) is a subring it is sufficient to show that if x and y are elements of 1(a) then x y and xy are elements of 1(a). Suppose x and y are in 1(a). Then, using Theorem 2.11 and Theorem 2.5, 0 < jx yÂ¡ A |a| < (|x| + |y|) A Â¡aj < (Â¡xi A |a|) + (|y i A Â¡aÂ¡) = 0. Hence x y is in 1(a). And Corollary - 23 - 4.1 asserts that xy is in J_(a). If x V y is also an element of j_(a) then 1(a) will be a sublattice. By Theorem 2.10, 0 < jx V y| A |aÂ¡ <(|xÂ¡ V |y|) A |a( = (ix| A Â¡a|) V (|y| A |a|) =0 V 0 = 0, Thus 1(a) is a dl-subring. Theorem 4.8. 1(a) is a ring ideal. Proof. It follows directly from Corollary 4.1 that 1(a) is closed under both left and right multiplica tion by arbitrary elements of R. Theorem 4.9,. 1(a) is an rl-ideal. Proof. 1(a) is obviously a normal subgroup. If x is an element of 1(a) and y is an element of R such that |y| < |x| then 0 < ja| A |yÂ¡ < Â¡aÂ¡ A |xj =0, and thus y is in 1(a). Since by Theorem 4.8 1(a) is already a ring ideal, it is an rl-ldeal. Theorem 4.10. If a > 0 and b > 0, l(a + b) = 1(a) r\ 1(b). Proof. If x Is in l(a + b) then 0 = |x| A |a + b| > |xÂ¡ A |a| > 0. Sox belongs to 1(a). Similarly x belongs to 1(b), and we have l(a + b) Q 1(a) r\ 1(b). Now suppose x is in both 1(a) and 1(b). Then by Theorem 2*5 and Theorem 2.11, 0 < |xÂ¡ A |a+bÂ¡ <(|x| A Â¡a|) = (|x| A Â¡bi) = 0. Thus the theorem is established. Theorem 4.11. If a > 0 and b > 0, l(a V b) = 1(a) n 1(b). - 24 - Proof. Suppose x belongs to j_(a V b). Then 0 = |x| A |a V b| = (|xÂ¡ A a) V (|x| A b). Since 1x| A a > 0 and |x| A b > 0 we have that |x| A a = 0 and |x| A b = 0. And [(a V b) C 1(a) nl(b), On the other hand let x be in j.(a) r\ J.(b). Then 0 < |xÂ¡ A |a V b| < |x A a| V |xAb| = 0, and the proof is complete. Theorem 4-.12. For a > 0, Â¡ax| =a|x|, ax+ = (ax)+, and ax" = (ax)". If a < 0, |ax| = -a|x|, ax+ = (ax)", and ax- = (ax)+. Proof. Take a > 0. Â¡ax| = ax V -ax = a(x V -x) = a|xj. ax+ = a(x VO) = ax V 0 = (ax)+. ax- = a(x A 0) = ax A 0 = (ax)". Nov: take a < 0. jax| = ax V -ax = -a(-x V x) = -a|x|. ax+ =a(xV0) = ax A 0 = (ax)", ax =a(xA0) = ax V 0 = (ax)+. Theorem 4.13. If e is an ace, ex < e|x| < jx|, and exv < (ex)v. Proof, ex < |ex| = ejxj < 1|xÂ¡ = |x|. exv = e(x VI) = ex V e < ex V 1 = (ex)V. Theorem 4.l4. Let X be a subset of R containing every x such that ax = xa for every a in R. X is a dl-subring. Proof. Suppose x and y are in X. Then (x y)a = xa ya = ax ay = a(x y) for every a in R. And xya = xay = axy for every a. Hence X is a subring. - 25 - Further (x V y) a = (x V y)(a+ + a~) = (xa+ V ya+) + (xa~ A ya) = (a+x V a+y) + (ax A ay) = a+(x V y) + a(x V y) = a(x V y). Thus X is a dl-subring. Definition 4.5. In a di-ring with a unit, where e is an ace, J(e) will denote a subset containing every element x of H such that x = ea = be for some a and b in R. Theorem 4.15. J(e) is a dl-subring. Proof. Suppose x = ea = be and y = ec = de are elements of J(e). Then x y = ea ec = e(a c) and x- y = be de = (b d)e. Also xy = eade = e(ade) = (ead)e. So x y and xy are in J(e) and J(e) is a subring. Also xV y = ea V ec = e(a V c) and x V y = be V de = (b V d)e. Thus x V y is in R. Theorem 4.l6. If e-j_ and eg are aces and e^ A e^ = 0 then J(ep) AAJieg) = {0} (the set consisting only of the element 0) . Proof. 0 is an element of J(e^) since 0 = e^O = Oe^. Similarly 0 is in J(e2). Suppose that at least one of the aces is 0, say e-^ = 0. The theorem is trivially true since J(e^) = {0}. Consider the case where e^ f 0 and e^ f 0. Suppose an element x ^ 0 is in J(e^) ^ J(g). Then x = e^a = egb for some a and b in R. Then, multiply ing by e^, e.^a = eie2t)* Theorem 2.10 e.^ = ^ and by Theorem 2.12 e^eg ~ * So ela = = Hence x = e^a = 0, contrary to assumption. BIBLIOGRAPHY 1. Garrett Blrkhoff, Lattice Theory. (revised edition), Am. Math. Soc., New York, 1942>. 2. Garrett Blrkhoff, Lattice-ordered groups. Annals of Math. 43 (1942), pp. 298-331. 3. N. Bourbakl, Algebre. Actuantes Scientifiques et Industrlelles 1179, Paris, 1952. 4. Hans Hermes, ElnfUhrung In die Verbandstheorie. Berlin, 1955. 5. Hildegoro Nakano, Modern Spectral Theory. Tokyo, 1950. 6. B. L. van der Waerden, Modem Algebra. Volume I, New York, 1949. - 26 - BIOGRAPHICAL SKETCH Mary Muskoff Neff was bom January 20, 193> in Jacksonville, Florida. She attended Purdue University from 19^7 to 1952 and belonged to Delta Bho Kappa, scho lastic honorary fraternity. She received a Bachelor of Science degree from Purdue in 1951. The following year she was a teaching assistant at Purdue, and in 1952 re ceived the Master of Science degree there. After a year at Bell Telephone Laboratories, Murray Hill, New Jersey, she came to the University of Florida. She was a teaching assistant (1953 195*0 > a graduate fellow (195^ 1955) and an instructor (1955 - 1956) in the Department of Mathematics. - 27 - This dissertation was prepared under the direction of the chairman of the candidates supervisory committee and has been approved by all members of the committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 11, 1956. Dean, College of Arts and Sciences Dean, Graduate School SUPERVISORY COMMITTEE: Chairman Co7?haSai^ MhP Â£A - 24 - Proof. Suppose x belongs to j_(a V b). Then 0 = |x| A |a V b| = (|xÂ¡ A a) V (|x| A b). Since 1x| A a > 0 and |x| A b > 0 we have that |x| A a = 0 and |x| A b = 0. And [(a V b) C 1(a) nl(b), On the other hand let x be in j.(a) r\ J.(b). Then 0 < |xÂ¡ A |a V b| < |x A a| V |xAb| = 0, and the proof is complete. Theorem 4-.12. For a > 0, Â¡ax| =a|x|, ax+ = (ax)+, and ax" = (ax)". If a < 0, |ax| = -a|x|, ax+ = (ax)", and ax- = (ax)+. Proof. Take a > 0. Â¡ax| = ax V -ax = a(x V -x) = a|xj. ax+ = a(x VO) = ax V 0 = (ax)+. ax- = a(x A 0) = ax A 0 = (ax)". Nov: take a < 0. jax| = ax V -ax = -a(-x V x) = -a|x|. ax+ =a(xV0) = ax A 0 = (ax)", ax =a(xA0) = ax V 0 = (ax)+. Theorem 4.13. If e is an ace, ex < e|x| < jx|, and exv < (ex)v. Proof, ex < |ex| = ejxj < 1|xÂ¡ = |x|. exv = e(x VI) = ex V e < ex V 1 = (ex)V. Theorem 4.l4. Let X be a subset of R containing every x such that ax = xa for every a in R. X is a dl-subring. Proof. Suppose x and y are in X. Then (x y)a = xa ya = ax ay = a(x y) for every a in R. And xya = xay = axy for every a. Hence X is a subring. - 25 - Further (x V y) a = (x V y)(a+ + a~) = (xa+ V ya+) + (xa~ A ya) = (a+x V a+y) + (ax A ay) = a+(x V y) + a(x V y) = a(x V y). Thus X is a dl-subring. Definition 4.5. In a di-ring with a unit, where e is an ace, J(e) will denote a subset containing every element x of H such that x = ea = be for some a and b in R. Theorem 4.15. J(e) is a dl-subring. Proof. Suppose x = ea = be and y = ec = de are elements of J(e). Then x y = ea ec = e(a c) and x- y = be de = (b d)e. Also xy = eade = e(ade) = (ead)e. So x y and xy are in J(e) and J(e) is a subring. Also xV y = ea V ec = e(a V c) and x V y = be V de = (b V d)e. Thus x V y is in R. Theorem 4.l6. If e-j_ and eg are aces and e^ A e^ = 0 then J(ep) AAJieg) = {0} (the set consisting only of the element 0) . Proof. 0 is an element of J(e^) since 0 = e^O = Oe^. Similarly 0 is in J(e2). Suppose that at least one of the aces is 0, say e-^ = 0. The theorem is trivially true since J(e^) = {0}. Consider the case where e^ f 0 and e^ f 0. Suppose an element x ^ 0 is in J(e^) ^ J(g). Then x = e^a = egb for some a and b in R. Then, multiply ing by e^, e.^a = eie2t)* Theorem 2.10 e.^ = ^ and by Theorem 2.12 e^eg ~ * So ela = = Hence x = e^a = 0, contrary to assumption. - 22 an ordinary pair. Using Theorem 3*5 we have 0 < |ab| A Â¡ba| < 1 aj |b1 A |b)|a| < ( | a | V |b[)(Â¡a| A |b|) =0. Thus |ab| A Â¡ba| = 0. Theorem 4.6. If a and b are orthogonal then for every x and y in R, ax 1 by and xa 1 yb. Proof. Using Theorem 3*5 0 < Â¡ax| A j by | < I a||x| A j b| |y| < ia|(|xi V |yÂ¡) A |b|(|x| V |y|) = ( J a! A Â¡b j ) ( | x| V | y | ) =0. Hence ax J_ by. Similarly xa 1 yb. Corollary iui. If a and b are members of a dl-ring with a unit and a and b are orthogonal, then for every x in R,a 1 bx and a J. xb. Definition 4.3. A subset S of a dl-ring R is called a dl-subrlng if S is a dl-ring and if it is a sublattice of R viewed as a lattice. Definition 4,4. A subset of a dl-ring R with a unit that contains every element x of R such that x is orthogonal to the element a is denoted by 1(a) Theorem 4.7. 1(a) is a dl-subring. Proof. To show that 1(a) is a subring it is sufficient to show that if x and y are elements of 1(a) then x y and xy are elements of 1(a). Suppose x and y are in 1(a). Then, using Theorem 2.11 and Theorem 2.5, 0 < jx yÂ¡ A |a| < (|x| + |y|) A Â¡aj < (Â¡xi A |a|) + (|y i A Â¡aÂ¡) = 0. Hence x y is in 1(a). And Corollary - 23 - 4.1 asserts that xy is in J_(a). If x V y is also an element of j_(a) then 1(a) will be a sublattice. By Theorem 2.10, 0 < jx V y| A |aÂ¡ <(|xÂ¡ V |y|) A |a( = (ix| A Â¡a|) V (|y| A |a|) =0 V 0 = 0, Thus 1(a) is a dl-subring. Theorem 4.8. 1(a) is a ring ideal. Proof. It follows directly from Corollary 4.1 that 1(a) is closed under both left and right multiplica tion by arbitrary elements of R. Theorem 4.9,. 1(a) is an rl-ideal. Proof. 1(a) is obviously a normal subgroup. If x is an element of 1(a) and y is an element of R such that |y| < |x| then 0 < ja| A |yÂ¡ < Â¡aÂ¡ A |xj =0, and thus y is in 1(a). Since by Theorem 4.8 1(a) is already a ring ideal, it is an rl-ldeal. Theorem 4.10. If a > 0 and b > 0, l(a + b) = 1(a) r\ 1(b). Proof. If x Is in l(a + b) then 0 = |x| A |a + b| > |xÂ¡ A |a| > 0. Sox belongs to 1(a). Similarly x belongs to 1(b), and we have l(a + b) Q 1(a) r\ 1(b). Now suppose x is in both 1(a) and 1(b). Then by Theorem 2*5 and Theorem 2.11, 0 < |xÂ¡ A |a+bÂ¡ <(|x| A Â¡a|) = (|x| A Â¡bi) = 0. Thus the theorem is established. Theorem 4.11. If a > 0 and b > 0, l(a V b) = 1(a) n 1(b). 2 lattice-ordered groups are used in obtaining the results on lattice-ordered rings. In Chapter III a lattice-ordered ring is defined following the manner of Garrett Birkhoff. The results of Chapter III and Chapter IV are believed to be new. belonging to R is defined, is a ring provided (1) R is a commutative group with respect to the operation + . (2) a(bc) = (ab)c for all a, b, c in R. (3) a(b + c) = ab + ac and (b + c)a = ba + ca for all a, b, c in R. Definition 1.22. A subset S of a ring R is a subring of R if S forms a ring under the operations of R. Theorem 1.12. A subset S of a ring R is a subring if and only if for a, b in S, a b and ab are in S. Definition 1.23. A nonempty subset M of a ring R is an ideal if a and b in M imply a b is in M and for a in M and r in R, ar and ra are in M, Theorem 1.13. An ideal is a normal subgroup of R regarded as a commutative group. The congruence relations on a ring R are the partitions of R by the oosets of its different ideals. - 12 Proof. Expanding by Theorem 2.S, |(a V x) (b V x)| =(aVxVbVx)- C(a V x) A (b V x)] = ( a V b V x) - [( a A b) V (x A b) V (a A x) V x] = (a V b V x) - [(a A b) V x]. Similarly Â¡a b| = (a V b) (a A b). Now let a V b = y and a A b = y*, and let y y* = t (note that t > 0). Then y V x = (t + y*) V x = t + [y* v (-t + x) ] < t + (y* V x). And (y V x} (y* V x) < t. Returning to the original notation this yields (a V b V x)- [(a A b)V x] < (a V b) (a A b), or |(a V x) (bVx)| < |a b|. Theorem 2.10. |aVb| <|aÂ¡ V |b|; |aAb| * < |aj A Â¡b1. Proof. |aVb| =aVbV -(a V b) = (a V b V -a) A (a V b V -b) < (a V b V -a V -b) A (a V b V -b V -a) = |a| V |bi. The second part of the theorem follows similarly. Theorem 2.11. In a commutative 1-group I a + b | < | a | + | b ( . Proof. Using Corollary 2.3 and Theorems 2.4 and 2.6, |a + bÂ¡ = (a + b)+ (a + b)~ < a+ + b+ a" b = I a. | + | b | . Definition 2.4. a and b are said to be orthogonal, in symbols a J_ b, provided |a| A |b| =0. Theorem 2.12. If a j. b then |a| + |b| = |b| + Â¡a| . Proof. This follows immediately from Theorem 2.2 since b V a = a V b. ON LATTICE-ORDERED RINGS By MARY MUSKOFF NEFF A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1956 -li the 1-ring such that e V e* = 1 and e A e =0. Theorem 3.g. If e is an ace and if e-^ V ej = e and e-j. A e2 = 0, then e-j. is an ace and e-^' =62 + 6*. Proof. 0 < e2 A e1 < e A e* =0, so that e2 + e' = (e2 V e*) + (e2 Ae1) = e2 V e*. Then e^ V e-^' = e^ V (e2 V e') = e V e' =1. In addition, 0 < e-^ A e^' = A (e2 V e') = (e^ A e2) V (e-^ A e') = 0 V (e^ A e1) < e a e' =0. Thus e1 is an ace. Theorem 3.9. The set of aces in an 1-ring form a Boolean algebra. Proof. If e^ and e2 are aces then e2 V e2 is an ace, for (e^ V e2)' = e^ A e2'. Dually e^ A e2 is an ace. Theorem 3.10. For any ace e(i)e+e'=l; (ii)ee' =0; and (iii) e2 = e. Proof. (i) e+ e' = (e V e') + (e A e') =1. (ii) ee' < e V ee* = el V ee' < e(l V e') = el = e. Similarly ee1 < e*. So 0 < ee' < e A e' = 0, (iii) e = e(e + e') = e2 + ee' = e2. Theorem 3.11. If e is an ace then (i) ex+ > (ex)+; (ii) ex" < (ex)"; (iii) ex^ < (ex)A; and (iv) ex J_ x ex. Proof, (i) ex+ = e(x V 0) > ex V 0 = (ex)+; (ii) ex = e(x A0) < ex A 0 = (ex); (iii) ex^ =e(x A 1) < ex A e < ex A 1= (ex)A; and CHAPTER IV DISTRIBUTIVE LATTICE-ORDERED RINGS Definition 4.1. A set of elements R will be called a distributive lattice-ordered ringr or a dl-rlngr if (1) R is an 1-ring. (2) a+(b V c) = a+b V a+c and (b V c)a+ = ba+ V ca+ for all a, b, c In R. Consider the ring of n-dimensional vectors where a sum is defined in the usual fashion, the product of two vectors (a^ a2, ***, an) and (b^, b2> bn) is defined as (a1b1, a2b2, ***, a^t^) and for i = 1, n the set of i-th components is a simply-ordered ring. If we lattice- order this ring by letting (a^, a2, **, a^) > 0 mean aj_ > 0 for each i, then the system forms a dl-rlng. On the other hand the ring of n x n matrices A = [a^], ordered by letting A > 0 mean every a >0, forms an 1-ring but not J. J a dl-ring. The theorems of Chapter IV pertain to dl-rings and their elements. Theorem 4.1. a+(b A c) = a+b A a+c for all a, b, c In a dl-ring. Proof. a+(b A c) =-a+(-b V -c) = -(-a+b V a+c) = a+b A a+c. 20 - This dissertation was prepared under the direction of the chairman of the candidates supervisory committee and has been approved by all members of the committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 11, 1956. Dean, College of Arts and Sciences Dean, Graduate School SUPERVISORY COMMITTEE: Chairman Co7?haSai^ MhP Â£A - 11 - Proof, (a A b) + (a A c) = (a + a) A (b + a) A (a + c) A (b + c). Since a, b, c are positive a + a > a, b + a > a, and a + c > a. Then (a A b) + (a A c) > a A a A a A (b+c) = a A (b+c). Theorem 2.6. If at least one of the elements a, b, c is negative then a A (b + c) > (a A b) + (a A c). Proof, (a A b) + (a A c) = (a + a) A (a + c) A (b + a) A (b + c). If a < 0 then a + a < a and (a A b) + (a A c) < a A (b+c). Similarly the inequality holds if either b or c is negative. Lemma 2.1. (-a)+ = -(a"). Proof, (-a)+ = -a V 0 = -(a A 0) = -(a-). Theorem 2*1. |a| > 0. Proof. Obviously a V -a > a A -a. Then (a V -a) (a A -a) = (a V -a) + (a V -a) > 0. But [(a V -a) A 0] + [(a V -a) A 0] = {[(a V -a) A 0] + (aV -a)} A {( a V -a) A 0} = [(a V -a) + (a V -a)] A (a V -a) A 0 = (a V -a) AO. Subtraction of (a V -a) A 0 from both sides of the equation yields (a V -a) A 0 = 0. Hence I a| = a V -a > 0. Theorem 2.S. |a b| = (a V b) (a A b). Proof, (a V b) (a A b) = [(a V b) a] V [(a V b) b] = 0 V b a V a b V 0 = Â¡a b|. Corollary 2.3. |a| = a+ a". Theorem 2.9. |(a V x) (b V x)| < |a b|. - 19 - Theorem 3.15. The rl-ideals of an 1-ring R form a complete distributive lattice, as do the congruence relations on R. Proof. It is obvious that the congruence relations on an 1-ring R are a subset of the congruence relations on R regarded purely as a lattice. That this subset is a complete sublattice is shown by Theorem 1.3, and this sublattice is distributive according to Theorem 1.4, CHAPTER I PRELIMINARY CONCEPTS Definition 1.1. An algebra A Is a set of elements together with an arbitrary number of operations fa. Each fa shall be a single-valued function assigning for a finite n = n(a), to every sequence (x^,***,xn) of n ele ments of A, a value f^ix^,* *,x^) in A. Definition 1.2. An equivalence relation is a binary relation = satisfying (1) For all x, x = x. (Reflexive) (2) If x = y, then y = x. (Symmetric) (3) If x = y and y = z, then x = z. (Transitive) Definition 1.1. A congruence relation on an algebra A Is an equivalence relation having the substitu tion property for each fa. In other words, if x^ = y^isO (i = 1, , n) then f^x^ ,xn) = fa(y1, * ,yn) (). Definition 1.4. A partly ordered set is a set with a binary relation < satisfying (1) For all x, x < x. (Reflexive) (2) If x < y and y < x, then x = y. (Antisymmetric) (3) If x < y and y < z, then x < z. (Transitive) Definition 1.5. If for every x and y in a partly ordered set either x < y or y < x, then the set is a chain or a simply ordered set. - 3 - - 7 - Definition 1.17. A group is said to be commutative if a + b = b + a. Theorem l.g. -(a + b) = -b a. [-b a is a short notation for -b + (-a).] Definition l.lg. A subset S of a group G is a subgroup if it is itself a group with respect to the oper ation of G. Theorem 1*1- A subset S of a group G is a sub group if and only if for any two elements a, b of S, a b is in S. Definition 1.19. If S is a subgroup of G and a is an element of G the set a + S (the set of elements a + s where s is in S) is a left coset. Similarly S + a is a right coset. A set that is both a left coset and a right coset is called a coset. Theorem 1.10. a + S and b + S are either identical or disjoint. Hence the various cosets of a group G with respect to a subgroup S constitute a partition of G. Definition 1.20. A normal subgroup N of a group G is a subgroup where a + N N + a for every a in G. Theorem 1.11. The congruence relations on a group G are the partitions of G by the cosets of its different normal subgroups. Definition 1.21. A set of elements R, in which for any two elements a, b, a sum a + b and a product ab - IS - in M. Hence M is a convex 1-subring. Theorem 1 ,l4. The congruence relations on an 1-ring R are the partitions of R into the cosets of its rl-ideals. Proof. It was pointed out in Chapter I that the congruence relations on a ring are the partitions of the ring into the cosets of its different ideals. The congru ence relations on an 1-ring R are those having the substi tution property for the lattice operations of meet and join as well as for addition and multiplication. Recall the fact that a V b = (a b)+ + b and that a A b = -(-a V -b). If an equivalence relation Q has the substitution property for addition and multiplication then it will have it for joins and meets providing merely that a = b(<9) implies a+ = b+(0). Let M be an rl-ideal of R. If a b is in M then, since |a+ b+| = |(a V 0) (b V 0)| < Â¡a-b|, a+ b+ is in M. In other words a = b mod M implies a+ = mod M. Conversely suppose M is the set of elements congruent to 0 under a congruence relation O. Then M is an ideal (ring) of R. And since M is evidently a normal subgroup of R then we have only to show that if a is in M and W < I a I then x is in M. |x| < |a| implies that aA-a completes the proof. - 13 - Definition 2.5. An 1-ldeal of an 1-group G is a normal subgroup of G that contains with any a all x such that |x| < |a|. Theorem 2.15. The congruence relations on any 1-group are the partitions of G into the cosets of its different 1-ideals. (Proof omitted. See [1, p. 222]) - 6 - Theorem 1.5. Let L be the lattice of congruence relations on an algebra A. If B is obtained from A by- introducing further finitary operations then the lattice of congruence relations on B is a sublattice of L. Definition 1.15. A lattice is distributive if its elements satisfy x A (y V z) = (x A y) V (x A z). Theorem 1.6. A distributive lattice satisfies x V (y A z) = (x V y) A (x V z). Theorem 1.7. A lattice is distributive if and only ifxVy=xVz and x A y = x A z imply y = z. Definition l.l4. A Boolean algebra may be defined as a complemented distributive lattice. Definition 1.15. By the set-theoretic union. XU Y, of two sets X and Y, is meant a set consisting of all the elements in either X or Y. The set-theoretic Intersection,. X A Y, is the set consisting of elements common to X and Y, Definition l.lo. A non-empty set G of elements, together with an operation which assigns to every pair of elements a, b of G a unique value in G denoted a + b, is a group if the following postulates are fulfilled: (1) For all a, b, c, a + (b + c) = (a + b) + c. (2) There exists an element 0 in G such that 0 + a = a + 0 = a. (3) If a is in G there exists an element -a in G such that a + (-a) = -a + a = 0 CHAPTER III LATTICE-ORDERED RINGS Definition 3.1. A set of elements R will be called a lattice-ordered ring, or an l-rlngf if It satisfies the following postulates: (1) R is a ring with respect to addition and multipli cation. (2) R is an 1-group under addition. (3) The product of any two positive elements is positive. Since an 1-ring is also a commutative 1-group the results obtained for such 1-groups, some of which are out lined in the preceding chapter, will be valid for 1-rings. It is of interest to discuss relations involving the multi plication operation in addition to the lattice and group operations. The theorems of Chapter III refer to the elements of an 1-ring. Theorem 1*1. The product of two negative elements is positive; the product of a positive and a negative ele ment is negative. Proof. First, if a < 0 and b < 0 then -a > 0 and -b > 0. Thus ab = (-a)(-b) >0. Second, if a < 0 and b > 0 then -a > 0 and -ab >0 or ab < 0. Theorem 3.2. a > 0 and b > c imply ab > ac; a < 0 - 14 .- INTRODUCTION In modern mathematics an algebra usually means a set of elements together with a number, perhaps infinite, of single-valued finitary operations. A lattice, a type of partly ordered set, is usually classified as an algebra, as is a ring. A lattice-ordered ring, or 1-ring, is an algebra which is both a ring and a lattice, having then four oper ationsthe two ring operations and the two lattice oper ations. In addition some specified relations involving a mixture of lattice and ring operations must hold. Rings have been studied extensively for many years, and for the last twenty years a steady flow of material concerning lattices has appeared. However the mathemati cal literature appears devoid of work on structures which are both lattices and rings, except for the case where the lattice is a chain. This work is concerned with some algebraic prop erties of 1-rings, Chapter I contains preliminary mate rial chosen for its bearing on the later chapters. Def initions and theorems are given, with proofs generally omitted. Chapter II, which deals with lattice-ordered groups, is presented since a lattice-ordered ring is also a lattice-ordered group. Also many of the properties of 1 BIBLIOGRAPHY 1. Garrett Blrkhoff, Lattice Theory. (revised edition), Am. Math. Soc., New York, 1942>. 2. Garrett Blrkhoff, Lattice-ordered groups. Annals of Math. 43 (1942), pp. 298-331. 3. N. Bourbakl, Algebre. Actuantes Scientifiques et Industrlelles 1179, Paris, 1952. 4. Hans Hermes, ElnfUhrung In die Verbandstheorie. Berlin, 1955. 5. Hildegoro Nakano, Modern Spectral Theory. Tokyo, 1950. 6. B. L. van der Waerden, Modem Algebra. Volume I, New York, 1949. - 26 - 21 Theorem 4.2. (i) a(b V c) = ab A ac; (11) a(b A c) = ab V ac. Proof. a~(b V c) = -(-a~)(b V c) = -(-ab V -ac) = ab A ac, and thus (i) is proved. A proof for (li) Is similar. Definition 4.2. Two elements a and b are said to form an ordinary pair if they satisfy the inequality ab A ba < (a V b)(a A b) < ab V ba. Theorem 4.1. If a and b are ordered they form an ordinary pair. Proof. Either a V b = a and a A b = b or a V b = b and a A b = a. Theorem 4.4. Every pair of positive elements (or of negative elements) forms an ordinary pair. Proof. If a and b are positive elements then (a V b)(a A b) = (a V b)a A (a V b)b > ba A ab. Also (a V b)(a A b) = a(a A b) V b(a A b) < ab V ba, and thus the theorem is established for positive elements. If a and b are negative elements (a V b)(a A b) = (a V b)a V (aVb)b=(aaA ba) V (ab A bb) < ba V ab. Also (a V b)(a A b) = a(a A b) A b(a A b) = (aa V ab) A (ba V bb) > ab A ba, and the proof is complete. Theorem If a and b are orthogonal then ab and ba are orthogonal. Proof. According to Theorem 4.4, |a| and |b| form ACKNOWLEDGMENTS 'The x^rlter wishes to extend her thanks to the members of her supervisory committee, and particularly to Professor R. G. Blake, who directed the preparation of this dissertation. - ii - TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii INTRODUCTION 1 Chapter I. PRELIMINARY CONCEPTS 3 II. LATTICE-ORDERED GROUPS 9 III. LATTICE-ORDERED RINGS l4- IV. DISTRIBUTIVE LATTICE-ORDERED RINGS .... 20 BIBLIOGRAPHY 26 BIOGRAPHICAL SKETCH 27 \ ill |