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Some Properties of the Lattice Oe All Equivalence

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Some Properties of the Lattice Oe All Equivalence SOME PROPERTIES OF THE LATTICE OE ALL EQUIVALENCE RELATIONS ON A FINITE SET DISSERTATION Presented in Partial Fulfillment of the Requirements for the degree Doctor of Philosophy in the Graduate School of The Ohio State University By Edley Wainright Martin, Jr., B. of E.E., B.A., M.A. The Ohio State University 1952 Approved hy: Adviser CONTENTS Acknowledgements. ..... ii I ITT BO DU GTI Oil..................................................... 1 CHAPTER I........................................................ E 1.1 The Simplicity of L(n)............................. 5 1.2 Quotient Lattices in L(n)............................8 1.3 Sublattice Generated by Three Element s...................... .15 CHAPTER II...................................................... 24 2.1 Some Elementary Generating Systems................24 2.2 Circles of Points........... 26 2.3 Converse to Theorem 2.2.3......................... 30 2.4 Generating Systems of Higher Dimension............36 2.5 Generating Systems with Fewer Elements............40 2.6 A System of Primitive Elements which Generate L(n)46 2.7 An Upper Bound for the Humber of Generators Necessary for L(n)....... ,,49 2.8 Remarks........................................... 51 Appendix........................................................ 54 Bibliography...................... 58 Autobiography. .............................................. ...59 918278 The author would like to express his sincere gratitude to his adviser, Professor Marshall Hell, Jr., whose counsel and inspiration was invaluable. The author would also like to express his appreciation to Professors Earl J. Mickle and Herbert J. Eyser for their most help­ ful suggestions. -1- SOME PROPERTIES OP THE LATTICE OP ALL EQUIVALENCE RELATIONS ON A FINITE SET INTRODUCTION Let S be a set with elements a, b, c......... A "binary relation rj which holds for certain pairs of elements of S will he called an equivalence relation if: (a) a f\j a. for all a in S; (h) a oj b implies b m a; (c) a oj h and b ro c implies a rj p. In cases where we are concerned with many such relations, we may write ar\jh(E^) to indicate that the particular relation Eg_ is under con­ sideration. A partition of the set S is a subdivision of the elements of the set S into disjoint subsets A (called blocks) so that S = 2.A,*.. cC The relationship a w b(E) if and only if a and b are in the same block of the partition P determines a one-one correspondence between the partitions of S and the set of all epuivalence relations on S. In the following the partition P and the equivalence relation E which correspond under this relationship will be considered to be different representations of the same thing, and the most convenient of the two notations will be used. Suppose Eq and E^ are equivalence relations on the set S. They may be partly ordered in the following way: Pq — % w^-enever arvb(EQ) then a.lso a oJb(E^). We may define E q fV as follows: arJb(Eg AE ^ ) if and. only if aoj UCEq) and a/vb(E^). The definition of Eq U E^ is more complicated: a rj b(EQ U E^) if and only if there exists a chain a = d^, d^, ... , d^., ^ + 1 = ^ elements of S such that dq /v d-^EQ or E^), d ^ or E ^ , ... , d^ro dk+1(EQ or E1). It can easily he shown that there then exists a subset dQ,d^, ... , d^ cLk+1 of these elements so that i) all of these are different, ii) dnu o j d,ii (E a\ ), where A = 0 or 1, and d. rj d.i.+ 1 (Er),^ where S 2 X + j(modulo 2). Such a reduced set of elements of S will he called a chain connecting a and h in Eq U E^, or sometimes an Eq U Eq chain connecting a and h. It can he shown that hoth Eq A Eq and Eq \J Eq are equivalence relations on S. Thus the set of all equivalence relations on (or partitions of) S form a lattice, which will he denoted L(S). ^ If S is a finite set consisting of n elements, then L(S) may he denoted L(n). It is well known that L(n) is a raatroid lattice (a relatively complemented semi-modular lattice) U ) . P. M. Whitman has shown [7 ] that every lattice is isomorphic with a suh-lattice of the lattice of all equivalence relations on some infinite set. We will he concerned only with lattices of equivalence relations on finite sets S, although a few of our results could obviously he extended to the case where S is infinite. Most of the above may he found in detail in [5 ) of the Bibliography. Hereafter numbers in brackets will refer to the corresponding reference in the Bibliography. -3- In Chapter I we will consider several rather diverse topics con­ cerned with properties of L(n). First we exhibit a new and more en­ lightening proof of the theorem ^5] that L(n) is simple. Then we consider homomorphisms and direct unions of lattices of partitions, applying the results obtained to characterize quotient lattices of L(n). Along this line there is a great opportunity for investigating more general homomorphism theory of lattices of partitions. It seems likely that a combination of Dilworth's general homomorphism theory and the theory of partitions might lead to some very interesting and important results. We conclude Chapter I with a partition representation (on a finite set) of a lattice which is generated by three elements and which has arbitrarily large dimension. This example is of special interest because it throws some light on one of the most important unsolved problems in the field of representations of lattices: For finite lattices can we omit the word "infinite" which is underlined in the statement of Whitman's result which we mentioned above? If we cannot it seems that difficulties should arise in representing large lattices generated by a small number of elements. However, this example shows that we can represent certain arbitrarily large finite lattices generated by only three elements by use of finite sets. The equivalence relations (or partitions) E^, ... , Er will be said to generate L(S) if every element of L(S) can be expressed - 4- as a lattice polynomial in E^, Eg, , E involving only a finite number of unions and intersections. In Chapter II we will consider some different generating systems for the lattice of all equivalence relations L(s) on a finite set S. Some of the results in Section 2.4 are inferior to the final results obtained. However, they are included in order to provide motivation and continuity. In Sections 2.2 and 2.3 a complete characterization is given of systems of lattice points which generate L(n). Our main result in Chapter II is that if n is less than or equal to the number of combinations of r things taken L at a time, then L(n) can be generated by r suitably chosen elements. Then we may ask: Will fewer than r elements serve to generate L(n)? The answer to this question is not known, and the question should make a good subject for future research. Last a word about the numbering of sections and theorems. Sections will be distinguished by two numbers, the first giving the chapter and the second giving the place of the section in the chapter. For example, 1.2 denotes the second section of Chapter II. For theorems and definitions we will use the same notation, adding a third number to give the position of the theorem in the designated section. Thus Theorem 1.2.3 is the third theorem in the second section of Chapter I. Lemmas of the Appendix will be denoted in order by Al, A2, and so forth. m 5« CHAPTER I 1.1 The Simplicity of L(n). A lattice is said to "be simple if it has no proper lattice-homo­ morphic images. It is well-known that L(n) is simple [5], However, a recent development ft?] in the theory of lattice homo morph is ms makes it possible to. improve considerably on Ore’s proof of this theorem. The following proof is not only much shorter, but also sheds a good deal more light on the actual situation. In the well-known theory of homomorphisms of modular lattices, the concept of projectivity is quite important. If a and b are elements of a lattice L, and a S b, then a/b denotes the lattice of all elements x of L such that a 2 x 2 b. a/b is called a quotient lattice, and the quotient is said to be prime if a covers b. The quotient a/b is said to be contained in the quotient c/d if c 2. a and b 2. d. Each of the quotients a/a q b and a u b/b is called a transpose of the other. Definition: 1.1.1 a/b and c/d are projective (denoted a/b c/d) if there exists a sequence of quotients a/b = x^/yp x2 ^ 2 * *** * xn ^ n = c/d such that x^/y^ is a transpose of x ^ ^ / y ^ p In a recent paper [3^ E. P. Dilworth introduced the concept of weak projectivity, and demonstrated that it is a fundamental concept in lattice homomorphism theory. Definition: 1.1.2 A quotient a/b is weakly projective into c/d (denoted a/b —» c/d) if there exists a sequence of quotients a/b = x^/y^, ... , xn/yn = c/d such that x^/y^ is contained in a transpose of xi+]/yi+p By considering a congruence relation (or homomorphism) as being S- determined "by a set of quotients which are collapsed, Dilworth arrives at the following corollary: L is simple if and only if for every pair of proper quotients a/b and c/d, there exists s chain from a to b whose consecutive quotients are all weakly projective into c/d.
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