Free Lattices. Department of Mathematics University of Hawaii Honolulu, Hawaii

Transcription

1 Free Lattices Ralph Freese Department of Mathematics University of Hawaii Honolulu, Hawaii Jaroslav Ježek MFF, Charles University Sokolovská Praha 8 Czech Republic jezek@@karlin.mff.cuni.cz J. B. Nation Department of Mathematics University of Hawaii Honolulu, Hawaii jb@@math.hawaii.edu

2 1991 Mathematics Subject Classification. Primary 06B25; Secondary 06 04, 68Q25, 06A06, 06B20 This research supported by NSF Grants DMS and DMS

3 Contents Introduction 1 Chapter I. Whitman s Solution to the Word Problem 9 1. Basic Concepts from Lattice Theory 9 2. Free Lattices Canonical Form Continuity Fixed Point Free Polynomials and Incompleteness Sublattices of Free Lattices 30 Chapter II. Bounded Homomorphisms and Related Concepts Bounded Homomorphisms Continuity Doubling and Congruences on a Finite Lattice A Refinement of the D Relation Semidistributive Lattices Splitting Lattices Day s Theorem: Free Lattices are Weakly Atomic Applications to Congruence Varieties 85 Chapter III. Covers in Free Lattices Elementary Theorems on Covers in FL(X) J closed Sets and the Standard Epimorphism Finite Lower Bounded Lattices The Lattice L (w) Syntactic Algorithms Examples Connected Components and the Bottom of FL(X) 115 Chapter IV. Day s Theorem Revisited 119 Chapter V. Sublattices of Free Lattices and Projective Lattices Projective Lattices The Free Lattice Generated by an Ordered Set Finite Sublattices of Free Lattices Related Topics 166 iii

4 iv CONTENTS 5. Finite Subdirectly Irreducible Sublattices of Free Lattices Summary 173

5 CONTENTS v Chapter VI. Totally Atomic Elements Characterization Canonical Form of Kappa of a Totally Atomic Element The Role of Totally Atomic Elements 192 Chapter VII. Finite Intervals and Connected Components Chains of Covers Finite Intervals Three Element Intervals Connected Components 212 Chapter VIII. Singular and Semisingular Elements Semisingular Elements Singular Elements 233 Chapter IX. Tschantz s Theorem and Maximal Chains Tschantz s Theorem Coverless Elements Maximal Chains 244 Chapter X. Infinite Intervals Tschantz Triples Join Irreducible Elements that are not Canonical Joinands Splittings of a Free Lattice 264 Chapter XI. Term Rewrite Systems and Varieties of Lattices Term Rewrite Systems No AC TRS for Lattice Theory An Extension The Variety Generated by L (w) A Lattice Variety with AC TRS More Varieties with AC TRS 295 Open Problems 301 Bibliography 303 List of Symbols 311 Index 313

6 Introduction This monograph covers the fascinating subject of free lattices. The arithmetic of free lattices is interesting and their structure is surprisingly complex. Although free lattices do have several nice properties, some reminiscent of properties of distributive lattices, discovering their nature is not easy. This book presents a thorough development of the structure of free lattices. A brief history. Although Richard Dedekind found the free modular lattice on three generators in 1900, Thoralf Skolem showed the word problem for finitely presented lattices had a uniform solution in 1920, and Garrett Birkhoff showed the free lattice on three generators was infinite, substantial insight into the nature of free lattices really began with the two papers [132, 133] of Philip Whitman which appeared in 1941 and In these he solved the word problem for free lattices and showed that there was a unique shortest term representing each element of the free lattice, known as the canonical form. He also established a strong connection between this syntactic notion and the arithmetic of the lattice. He showed that the canonical form is the best way to represent an element in a very natural lattice theoretical sense. Using this interplay, he was able to prove several strong results about free lattices; for example, the free lattice on a countably infinite generating set is embeddable in the one on a three element generating set. In the 1950 s and 1960 s, R. Dean, B. Jónsson, F. Galvin, and J. Kiefer obtained further structural results about free lattices. One of their primary interests was sublattices of free lattices, particularly finite sublattices. As part of his solution to the word problem, Whitman had shown that free lattices satisfy an important property now known as Whitman s condition, which states that the relation a b c d can hold in a free lattice only in one of the four trivial ways: a c d, b c d, a b c, or a b d. Jónsson showed that free lattices and all their sublattices satisfy a weak form of distributivity known as semidistributivity. As part of 1

7 2 INTRODUCTION their attack on the problem of characterizing finite sublattices of free lattices, Jónsson and Kiefer showed that if a lattice satisfies Whitman s condition and has a set of five elements, no four of which join above the fifth, then it is infinite. This implies that the breadth of a finite sublattice of a free lattice is at most four and that there is a nontrivial lattice equation true in all finite sublattices of a free lattice. Chapter I presents these and other classical results after first giving Whitman s solution to the word problem and other basic results. It also shows that free lattices are continuous, not complete, and that free lattices have a unary polynomial which is fixed point free. The first chapter is suitable as an introduction to free lattices in, say, a course on lattice theory or algebra. In the paper by Jónsson and Kiefer it was conjectured that a finite lattice could be embedded into a free lattice if and only if it satisfies Whitman s condition and is semidistributive. Jónsson worked on this conjecture, producing some unpublished notes that influenced further investigations by others. Ralph McKenzie s important paper [99] uses Jónsson s notes in the characterization of finite projective lattices. A great deal of work was put into Jónsson s conjecture, as it was usually termed. J. B. Nation finally confirmed the conjecture in a deep paper [104] appearing in Nation s proof is based on the approach developed in Jónsson s notes. This theorem appears in Chapter V, which also contains a characterization of projective lattices and other results on sublattices of free lattices. The notion of covering, one element being covered by another, denoted a b, naturally leads to several intriguing questions and plays an extremely important role in many of the deeper results on free lattices. Consequently, the theory of the covering relation in free lattices is well developed in this monograph. Whitman was the first to exhibit coverings in free lattices: he showed, for example, that the meet of all but one of the generators covers the meet of all of the generators. R. Dean, in unpublished work, showed that there were elements that covered no other element. When a b, we say a is covered by b and a is a lower cover of b. When a b (in any lattice), there is a unique maximal congruence relation, ψ(a, b), not containing a, b. McKenzie [99] showed that the homomorphic image of the free lattice corresponding to this congruence relation is a finite, subdirectly irreducible lattice with the additional property that each element of this lattice has a least and a greatest preimage in the free lattice, i.e., the natural homomorphism is bounded. These lattices, which are called splitting lattices, and the corresponding bounded homomorphisms give an effective tool for studying covers in

8 INTRODUCTION 3 free lattices. For example, with the aid of his techniques, McKenzie was able to show that deciding if a b holds in a free lattice is recursive. Chapter II gives a detailed coverage of bounded homomorphisms and splitting lattices. Jónsson s unpublished notes studied certain dependency relations in lattices. Although on the face it would not appear so, these relations are closely related to bounded homomorphisms. These and other relations are covered thoroughly in a general setting in this chapter. The chapter includes applications of these ideas and results to several areas of lattice theory and algebra, such as projective lattices, varieties of lattices, and the equational theory of congruence lattices of algebraic systems. The dependency relations of this chapter give an efficient representation of congruences on a finite lattice, facilitating fast algorithms for determining such properties as subdirect irreducibility and simplicity, which are developed in Chapter XI. In a significant paper [24], Alan Day, using his doubling construction, was able to prove that the variety generated by all splitting lattices was in fact all lattices. This implies that free lattices are weakly atomic: every nontrivial interval contains a covering. We present two proofs of this result; the first is in Chapter II. In Chapter IV we prove a somewhat stronger version of Day s Theorem, useful for other results on free lattices. Several basic questions about covers and about the structure of free lattices remained. Given an element, could one decide if it was covered by any other element? Could one effectively find all such covers? Is there a coverless element: one which covers no element and is covered by no element? How long could a chain of covers in a free lattice be? These questions were settled by Freese and Nation in a paper [62] which thoroughly developed the theory of covers in free lattices. A computationally efficient and theoretically useful algorithm was given which finds all elements covering a given element. Using this algorithm, it was possible to construct an element with no upper and no lower covers. In answering some of the deeper questions, two facts played a key role. Of course, if anelement a of a free lattice is join irreducible, it can coveratmostoneelement. Whenthiselementexists, wedenoteita. It turns out that a is almost always a proper meet and that its canonical expression has a unique meetand, denoted κ(a), not above a. This map κ is a bijection between the completely join irreducible elements and the completely meet irreducible elements. While making a table with some examples of elements and their κ s for an early version of [62], we observed that if the canonical expression for a was a 1 a n, then the canonical expression for a was usually just κ(a) a 1 a n. The

9 4 INTRODUCTION first key fact alluded to above was that the canonical expression for a is simply the meet of κ(a) and those a i s which are not above κ(a). At the time this seemed little more than a cute observation but turns out to play a crucial role in many of the deeper results on free lattices. Suppose that a = a 1 a n andsuppose that ahas bothanupper and a lower cover; say c a b. Since not every a i can be above b, by renumbering if necessary, one may assume that b a 1. The second key fact was that, in this situation, a 1 is totally atomic, i.e., each element properly above a 1 contains a cover of a 1, and the dual condition holds. Each free lattice has only finitely many totally atomic elements and this severely restricts chains of covers in free lattices. Indeed, except at the very top and bottom of free lattices, chains of covers can have length at most two. With the aid of these tools, it was possible to answer many of the open questions. For example, all lattices isomorphic to finite intervals were found. The basic theory of covers in free lattices is given in Chapter III. Chapter VI characterizes totally atomic elements and proves several important consequences of this characterization. Chapter VII uses the results of Chapters III and VI to derive some of the deeper results mentioned above on chains of covers, finite intervals, and connected components of the covering relation. A characterization of three element intervals, which is useful in the following chapters, is given. Perhaps because of the success of [62], there was little work on free lattices for a few years except for the characterization of the connected components of the covering relation in[51]. But one intriguing question remained. After characterizing all finite intervals of free lattices, Freese and Nation sought to show that every infinite interval in a free lattice contains the free lattice on a countably infinite generating set as a sublattice. By Whitman s work, it would be enough to show that every infinite interval contains a set of three elements which are join and meet irredundant. We felt proving this would be much easier than our work on finite intervals. We were quite surprised when we could not do this and, in fact, were not even able to show the existence of two incomparable elements. That is, we could not rule out the possibility of an infinite interval in a free lattice which is a chain! We were able to derive some unlikely consequences of such an interval, but none produced a contradiction. We showed this problem to Tom Harrison who derived much stronger (and more unlikely) consequences, but still without reaching a contradiction. Harrison showed it to Steve Tschantz, who, after working on it for some time, was able to show that such an interval could not exist, and, in fact, that every infinite interval contains the free lattice on ω generators. Interestingly, most of

10 INTRODUCTION 5 Tschantz s proof is devoted to showing that there is no infinite interval which is a chain. A new proof of Tschantz s Theorem is presented in Chapter IX. Given elements a < b in a free lattice, what are the possible types of maximal chains in the interval sublattice from a to b? In particular, can the ordered set of rationals between 0 and 1 be represented in this way? This question was posed informally to the first author by F. Galvin, J. Mycielski, and W. Taylor. Using Tschantz s Theorem, one can show that every infinite interval of a free lattice contains a coverless element, an element with no upper and no lower cover. In trying to construct a dense chain, we were led to semisingular elements. If a 1 a n is the canonical expression of an element a which is join irreducible and has a lower cover a, then a is semisingular if κ(a) a i, for some i. As we mentioned earlier, this situation is somewhat exceptional. We conjectured that, with a few trivial exceptions, every semisingular element must be the middle element of a three element interval. The truth of this would allow us to construct dense maximal chains in almost every infinite interval of a free lattice. However, the truth of our conjecture would also prove that no interval of a free lattice could be a chain and thus would substantially shorten the proof of Tschantz s Theorem. This made us doubtful the conjecture could be proved without a great deal of work, but we pressed on anyway and were pleased when we were able to prove it. This result is in Chapter VIII, and the applications to proving Tschantz s Theorem and to constructing maximal chains are in Chapter IX. In spite of these successes, Tschantz s (and our subsequent) techniques were not strong enough to answer the following question raised in [131]: can an interval have an isthmus? Specifically, are there elements a < c < b in a free lattice such that the interval from a to c and the one from c to b are both infinite and every element in the interval from a to b is comparable with c? After a sustained effort we were able to show that such an interval cannot exist. This result and some of its consequences are in Chapter X. Computing. In their early work on free lattices, Freese and Nation often had to put specific terms into canonical form. Whitman s algorithm for this, while efficient (even polynomial time when implemented correctly), is tedious for humans. Freese used the following alternate procedure: he would stare at the term, looking for any obvious reason it was not in canonical form. If he could not find any such reason, he would take the term to Nation, whose office is one floor higher, and ask him to look at it. Nation used a dual procedure. The inadequacy of

11 6 INTRODUCTION this procedure led us to develop computer programs to calculate in free lattices. These programs were later extended into a computer algebra system that deals with all types of lattices. Many of the examples in this monograph were found with the aid of this program. Several of the crucial lemmas were proved after first extensively searching for a counterexample with the program, and of course several conjectured results were disproved with this program. Chapter XI gives an analysis of many lattice theoretic algorithms. Our original intent was to only analyze those algorithms covered in this book: free lattice algorithms and algorithms for finite lattices which are related to free lattices such as determining if a finite lattice is a splitting lattice. Of course we had to develop the basic algorithms and structures for representing lattices, and as we proceeded, we found there is an interesting body of results spread over both the mathematics and computer science literature. We felt it would be useful to bring this knowledge together, so this chapter covers a much wider area than was originally planned. It covers the basic algorithms for ordered sets such as finding a linear extension. It discusses representations of lattices and proves the result of P. Goralčík, A. Goralčíková, V. Koubek, and V. Rödl [69] which shows that one can decide if an ordered set of size n is a lattice in time O(n 5/2 ). The dependence relations of Chapter II give efficient ways of representing and calculating in congruence lattices of finite lattices. For example, itisshownthatonecandecideifalatticeofsizenissimple, or is subdirectly irreducible, or is a splitting lattice, each in time O(n 2 ). The algorithm of K. Bogart and D. Magagnosc [13] to find a chain partition and a maximum sized antichain is presented, as well as an algorithm to find the lattice of all maximum sized antichains of an ordered set. We give a careful analysis of Whitman s algorithm for testing if a bandfor putting terms into canonical forminfreelattices, since all other free lattice algorithms use these. The basic algorithms for finitely presented lattices are also covered. Finally some remarks on computer algorithms for automatically drawing lattices are given, including our own new algorithm. Term rewrite systems were popularized by the work of D. Knuth and P. Bendix in [91]. Such systems give rewrite rules of a special form which transform any term into a normal form. In Chapter XII we show that, despite Whitman s very nice algorithm to put terms into canonical form, there is no associative, commutative term rewrite

12 INTRODUCTION 7 system for lattices. On the other hand we show that several varieties of lattices do have term rewrite systems. Several of the results of this monograph are new or at least have never been published. The existence of a fixed point free polynomial on the free lattice with three generators and the characterization of fixed point free slim polynomials in Chapter I is new. Many of the results of Chapters II and V are proved in a more general and more natural setting. Certain results in Chapters V, VI and VII are new, such as the example showing that there are infinitely many three element intervals in the free lattice on three generators in Chapter VII and the explicit formula for κ(w) of a totally atomic element in Chapter VI. All the results of Chapters VIII and X are new, as are the results on coverless elements and maximal chains in intervals of free lattices in Chapter IX. Several of the theorems and algorithms of Chapter XI are new, for example the O(n 2 log 2 n) time algorithm to recognize when a partially ordered set is semidistributive or is a splitting lattice, Theorem??. The O(n 2 ) algorithm for testing if a lattice is simple, or if it is subdirectly irreducible, Theorem??, is new. The results of Chapter XII are new. Courses. There are several ways to read this book. As we mentioned before, the first chapter can be used as an introduction to the classical results of free lattices. The first three sections of Chapter I are prerequisites for most of the rest of the book. Chapter II serves several purposes. It proves various results necessary for Chapters V and XII and some for Chapter III. It also develops several free lattice techniques which are useful in other areas of lattice theory and algebra, particularly varieties. Many free lattice results are proved in the more general setting of semidistributive lattices. The reader who is more interested in the structure of free lattices, in particular the theory of covers and its consequences, should read Chapter III and Chapters VI through X. Since Chapter II contains a great deal of material, most of which is not needed for Chapter III, we have written Chapter III so that it can be read without first reading Chapter II. Thus some of the results of Chapter II are reproved in the specialcaseoffreelattices, wheretheproofisofteneasier. Ontheother hand, the corresponding result from Chapter II is cited for the reader who has read it or who is interested in the more general formulation. The reader who is interested in sublattices of free lattices and in projective lattices should read Chapters II and V. The reader interested in the computational aspects of lattice theory should read Chapters XI and XII. Most sections of Chapter XI only require a basic knowledge of lattices. For the other sections the reader can refer back to the

13 8 INTRODUCTION earlier chapters as needed. Chapter XII requires Chapter I and parts of Chapter II. The monograph gives several open problems throughout the text. For convenience, they are repeated on page 301. We would like to thank Bjarni Jónsson, Stan Burris, and the referee for several suggestions. We would also like to thank our editor at the American Mathematical Society, Christine Thivierge, and the AMS technical support group for help with the preparation of the manuscript, which was typeset with AMS-L A TEX.

14 CHAPTER I Whitman s Solution to the Word Problem In this chapter, after first giving a brief introduction to lattice theory including Alan Day s doubling construction, we give the basic properties of free lattices including Whitman s solution to the word problem. Our proof is based on Day s doubling construction. Whitman s solution implies that the elements of a free lattice have a very nice canonical form and we investigate his solution and canonical form carefully. Several very interesting results about free lattice can be derived using this canonical form and the solution to the word problem alone. The existence of the canonical form easily implies that free lattices (and their sublattices) are semidistributive. We are also able at this point to prove some deeper theorems on free lattices. Some of these are well known such as Whitman s result that the free lattice with 3 generators has a sublattice isomorphic to the free lattice with a countable free generating set. Others are not well known. We prove Whitman s theorem that free lattices are continuous. We prove an unpublished result of the first author that there is a unary polynomial function on the free lattice with three generators without a fixed point. This is relevant to the problem of characterizing order polynomial complete lattices. Using this fixed point free polynomial, it is not hard to prove a little known fact of Whitman that the free lattice with 3 generators contains an ascending chain without a least upper bound. We also prove a result of Jónsson and Kiefer which shows that finite sublattices of free lattices have breadth at most four. This then implies that there is a nontrivial lattice equation which is satisfied by all finite sublattices of free lattices. 1. Basic Concepts from Lattice Theory We begin this section with a very brief review of the fundamental concepts associated with lattice theory. The reader can consult either Crawley and Dilworth s book [20] or McKenzie, McNulty, and Taylor s book [100] for more details. Other standard references are Birkhoff [10], Davey and Priestley [22], and Grätzer [73]. For the most part we follow the notation and terminology of [100]. We use 9

15 10 I. WHITMAN S SOLUTION TO THE WORD PROBLEM the notation A B for set inclusion, while A B denotes proper set inclusion. An order relation on a set P is a binary relation which satisfies (1) x x, for all x P (reflexivity) (2) x y and y x imply x = y, for all x, y P (anti-symmetry) (3) x y and y z imply x z, for all x, y, z P (transitivity) An ordered set (also known as a partially ordered set) is a pair P = P,, where P is a set and is an order relation on P. If P, is an ordered set, the relation is defined by x y if and only if y x. P, is an ordered set known as the dual of P,. Each concept and theorem about ordered sets has a dual obtained by reversing the roles of and. In proofs we often use the phrase by duality to express the symmetry between and. Of course x < y if and only if x y and x y, and > is defined dually. A relation which is reflexive and transitive, but not necessarily antisymmetric is called a quasiorder. If is a quasiorder on S, a b if a b and b a defines an equivalence relation on S. Then induces a natural partial order on P/. A chain C in an ordered set P is a subset of P such that any two elements of C are comparable, i.e., if x and y C then either x y or y x. An antichain is a subset A of P such that no two elements of A are comparable. The length of P is the supremum of the cardinalities of all chains in P, and the width is the supremum of the cardinalities of all antichains in P. Let S be a subset of P and a P. We say that a is the least upper bound of S if a is an upper bound for S, i.e., s a for all s S, and a b for any upper bound b of S. If it exists, we denote the least upper bound by S. The dual concept is called the greatest lower bound and is denoted by S. If S = {a,b} then S is denoted by a b and S by a b. The terms supremum and join are also used for the least upper bound and infimum and meet are used for the greatest lower bound. A lattice is an algebra L = L,,, with two binary operations which are both idempotent, commutative, and associative, and satisfy the absorptive laws: x (y x) = x and x (y x) = x for all x L. If L = L,, is a lattice, we can define an order on L by x y if and only if x y = x. Under this order, x y is the

16 1. BASIC CONCEPTS FROM LATTICE THEORY 11 greatest lower bound of x and y, and x y is the least upper bound of x and y. Conversely, an ordered set L, such that each pair of elements of L has both a greatest lower bound and a least upper bound defines a lattice. It is easy to see that the dual of a lattice L,, is L,,. An element a a lattice L is join irreducible if a = b c implies that either a = b or a = c. An element a is completely join irreducible if a = S implies a S. An element a is join prime if a b c implies that either a b or a c; it is completely join prime if a S implies a s for some s S. Naturally meet irreducible, completely meet irreducible, meet prime, and completely meet prime are defined dually. If a lattice has a least element, it is denoted by 0 and if it has a greatest element, it is denoted by 1. Note that, technically, 0 is join irreducible but not completely join irreducible. However, we will follow the long-standing convention that in a finite lattice, 0 is not regarded as join irreducible. (This is because we treat a finite lattice as a complete lattice.) Dually, in a finite lattice 1 is not considered to be meet irreducible. We let J(L) denote the join irreducible elements of L and M(L) denote the meet irreducible elements of L. If a < b are elements in a lattice L and there is no c L with a < c < b, then we say that a is covered by b, and we write a b. In this situation we also say that b covers a and write b a. In addition we say that b is an upper cover of a and that a is a lower cover of b. We also define a nameless equivalence relation on L by saying c is equivalent to d if there is a finite sequence c = c 0,c 1,...,c n = d such that c i either covers or is covered by c i+1. The blocks of the equivalence relation are called the connected components of the covering relation of L. The connected component of a L is the block containing a. If a b, then we let b/a denote the interval {x : a x b}. An order ideal in an ordered set P is a subset S of P such that whenever a b and b S then a S. An order filter is defined dually. A subset I of a lattice L is called an ideal if it is an order ideal and it is closed under finite joins. A filter is defined dually. If S L, then the ideal generated by S, the smallest ideal containing S, consists of all elements a L such that a s 1 s k for some s 1,...,s k S. Day s doubling construction. A useful construction for free lattice theory is Alan Day s doubling construction. We will use this construction in this chapter to derive one of the basic properties of free lattices, known as Whitman s condition, following Day s approach [26]. The doubling construction also plays a crucial role in the proof of Day s

17 12 I. WHITMAN S SOLUTION TO THE WORD PROBLEM important result [24] that free lattices are weakly atomic, discussed in Chapter II and Chapter IV. Let L be a lattice. A subset C of L is convex if whenever a and b are in C and a c b, then c C. Of course an interval of a lattice is a convex set as are lower and upper pseudo-intervals. A subset C of L is a lower pseudo-interval if it is a finite union of intervals, all with the same least element. An upper pseudo-interval is of course the dual concept. Let C be a convex subset of a lattice L and let L[C] be the disjoint union (L C) (C 2). Order L[C] by x y if one of the following holds. (1) x, y L C and x y holds in L, (2) x, y C 2 and x y holds in C 2, (3) x L C, y = u,i C 2, and x u holds in L, or (4) x = v,i C 2, y L C, and v y holds in L. There is a natural map λ from L[C] back onto L given by { x if x L C (1) λ(x) = v if x = v,i C 2. The next theorem shows that, under this order, L[C] is a lattice, denoted L[C]. Theorem 1.1. Let C be a convex subset of a lattice L. Then L[C] is a lattice and λ : L[C] L is a lattice epimorphism. Proof: Routine calculations show that L[C] is a partially ordered set. Let x i L C for i = 1,...,n and let u j,k j C 2 for j = 1,...,m. Let v = x i u j in L and let k = k j in 2. Of course, if m = 0, then k = 0. Then in L[C], { v if v L C (2) x 1 x n u 1,k 1 u m,k m = v,k if v C To see this let y be the right side of the above equation, i.e., let y = v if v L C and y = v,k if v C. It is easy to check that y is an upper bound for each x i and each u j,k j. Let z be another upper bound. First suppose z = a,r where a C. Since z is an upper bound, it follows from the definition of the ordering that a v and r k and this implies z y. Thus y is the least upper bound in this case. The case when z / C is even easier. The formula for meets is of course dual. Thus L[C] is a lattice and it follows from equation (2) and its dual that λ is a homomorphism which is clearly onto L.

18 2. FREE LATTICES Free Lattices Since the lattice operations are both associative, we define lattice termsover a set X, andtheir associatedlengths, inamanner analogous to the way they are defined for rings. Each element of X is a term of length 1. Terms of length 1 are called variables. If t 1,...,t n are terms of lengths k 1,...,k n, then (t 1... t n ) and (t 1 t n ) are terms with length 1+k 1 + +k n. When we write a term we usually omit the outermost parentheses. Notice that if x, y, and z X then x y z x (y z) (x y) z are all terms (which always represent the same element when interpreted into any lattice) but the length of x y z is 4, while the other two terms are both of length 5. Thus our length function gives preference to the first expression, i.e., it gives preference to expressions where unnecessary parentheses are removed. Also note that the length of a term (when it is written with the outside parentheses) is the number of variables, counting repetitions, plus the number of pairs of parentheses (i.e., the number of left parentheses). The length of a term is also called its rank. It is also useful to have a measure of the depth of a term. The complexity of a term t is 0 if t X, and if t = t 1 t n or t = t 1 t n, where n > 1, then the complexity of t is one more than the maximum of the complexities of t 1,...,t n. The set of subterms of a term t is defined in the usual way: if t is a variable then t is the only subterm of t and, if t = t 1... t n or t = t 1 t n, then the subterms of t consist of t together with the subterms of t 1,...,t n. By the phrase t(x 1,...,x n ) is a term we mean that t is a term and x 1,...,x n are (pairwise) distinct variables including all variables occurring in t. If t(x 1,...,x n ) is a term and L is a lattice, then t L denotes the interpretation of t in L, i.e., the induced n ary operation on L. If a 1,...,a n L, we will usually abbreviate t L (a 1,...,a n ) by t(a 1,...,a n ). Very often in the study of free lattices, we will be considering a lattice L with a specific generating set {x 1,...,x n }. In this case we will use t L to denote t L (x 1,...,x n ). If s(x 1,...,x n ) and t(x 1,...,x n ) are terms and L is a lattice in which s L = t L as functions, then we say the equation s t holds in L. Let F be a lattice and X F. We say that F is freely generated byx ifx generates Fandevery mapfromx into anylatticelextends to a lattice homomorphism of F into L. Since X generates F, such an extension is unique. It follows easily that if F 1 is freely generated by X 1 and F 2 is freely generated by X 2 and X 1 = X 2, then F 1 and F 2

19 14 I. WHITMAN S SOLUTION TO THE WORD PROBLEM are isomorphic. Thus, if X is a set, a lattice freely generated by X is uniqueuptoisomorphism. Wewill see thatsuch alatticealwaysexists. It is referred to as the free lattice over X and is denoted FL(X). If n is a cardinal number, FL(n) denotes a free lattice whose free generating set has size n. To construct FL(X), let T(X) be the set of all terms over X. T(X) can be viewed as an algebra with two binary operations. Define an equivalence relation on T(X) by s t if and only if the equation s t holds in all lattices. It is not difficult to verify that restricted to X is the equality relation, that is a congruence relation on T(X), and that T(X)/ is a lattice freely generated by X, provided we identify each element of x X with its class. This is the standard construction of free algebras; see, for example, [20] or [100]. This construction is much more useful if we have an effective procedure which determines, for arbitrary lattice terms s and t, if s t. The problem of finding such a procedure is informally known as the word problem for free lattices. In [132], Whitman gave an efficient solution to this word problem. Virtually all work on free lattices is based on his solution. If w FL(X), then w is an equivalence class of terms. Each term of this class is said to represent w and is called a representative of w. More generally, if L is a lattice generated by a set X, we say that a term t T(X) represents a L if t L = a. As Jónsson has shown in [81], certain basic aspects of Whitman s solution hold in every relatively free lattice and so it is worthwhile developing the theory in this more general context. A variety is a class of algebras (such as lattices) closed under the formation of homomorphic images, subalgebras, and direct products. A variety is called nontrivial if it contains an algebra with more than one element. By Birkhoff s Theorem [9], varieties are equational classes, i.e., they are defined by the equations they satisfy, see [100]. If V is a variety of lattices and X is a set, we denote the free algebra in V by F V (X) and refer to it as the relatively free lattice in V over X. If L is the variety of all lattices, then, in this notation, F L (X) = FL(X). However, because of tradition, we will use FL(X) to denote the free lattice. The relatively free lattice F V (X) can be constructed in the same way as FL(X). Notice that every nontrivial variety of lattices contains the two element lattice, which is denoted by 2.

20 2. FREE LATTICES 15 Lemma 1.2. Let V be a nontrivial variety of lattices and let F V (X) be the relatively free lattice in V over X. Then ( ) S T implies S T for each pair of finite subsets S, T X. Proof: Suppose that S and T are finite, disjoint subsets of X. As noted above, 2 V. Let f be the map from X to {0,1} which sends each x S to 1 and all other x s to 0. By the defining property of free algebras, f can be extended to a homomorphism from F V (X) onto 2, which we also denote by f. Then f( S) = 1 0 = f( T). This implies that S T, as desired. The next lemma will be used below to characterize when a subset of a lattice generates a free lattice. Lemma 1.3. Let a be an element of a lattice L generated by a set X. Suppose that for every finite subset S of X, ( ) a S implies a s for some s S. Then ( ) holds for all finite subsets of L. Proof: Let K be the collection of all sets U with X U L such that ( ) holds for every finite subset S of U. By hypothesis, X K. Suppose U K and that u and v U. Then U {u v} K. To see this, suppose that a S (u v) for some finite S U, but a s for all s S. Then, since a S u, ( ) implies a u. Similarly, a v and so a u v. It is trivial that U {u v} K. Since X generates L, we conclude that L K. Lemma 1.4. Let L be a lattice generated by a set X and let a L. Then (1) if a is join prime, then a = S for some finite subset S X, (2) if a is meet prime, then a = S for some finite subset S X. If X satisfies condition ( ) above, then (3) for every finite, nonempty subset S X, S is join prime and S is meet prime, (4) if X is the disjoint union of Y and Z, and F is the filter of L generated by Y and I is the ideal generated by Z, then L is the disjoint union of F and I. Proof: Since L is generated by X, every element of L can be represented by a term with variables in X. It follows from this and an easy inductiononthelengthofsuchatermthatif X = Y Z, then L = F I where F is the filter generated by Y and I is the ideal generated by Z.

21 16 I. WHITMAN S SOLUTION TO THE WORD PROBLEM To prove (1), let F be the filter generated by Y = {x X : a x} and let I be the ideal generated by Z = {x X : a x}. Since a is join prime, a / I, and so by the above observation, a F. This implies that a S, for some finite S Y. But every element of Y is above a; hence a = S, as desired. Of course (2) is proved dually. Let T be a finite, nonempty subset of X and let a = T. It follows from condition ( ) that condition ( ) holds for all finite subsets S of X. Hence, by Lemma 1.3, a is join prime. For (4), we have already observed that L = F I. If F I is nonempty, there would be subsets S and T of X with S T, contrary to our assumption. Combining Lemmas 1.2 and 1.4, we obtain the following corollary for relatively free lattices. Corollary 1.5. Let V be a nontrivial variety of lattices and let F V (X) be the relatively free lattice in V over X. For each finite, nonempty subset S of X, S is join prime and S is meet prime. In particular, every x X is both join and meet prime. Moreover, if x y for x and y X, then x = y. Unlike the situation for groups and Boolean algebras, the free generating set of a free lattice is uniquely determined, as the next corollary shows. Corollary 1.6. If L is a lattice generated by a set X which satisfies condition ( ), then the following hold. (1) If Y generates L then X Y. (2) Every automorphism of L is induced by a permutation of X. Inparticular, thesestatements holdfor the relativelyfreelattice, F V (X), for any nontrivial variety of lattices V. Moreover, the automorphism group of F V (X) is isomorphic to the full symmetric group on X. Proof: By Lemma 1.4(3), each x X is both join and meet irreducible. Hence if x is in the sublattice generated by Y, it must be in Y. This proves the first statement. The rest of the theorem follows easily from the first. The next corollary gives some of the basic coverings in free lattices, including the atoms and coatoms. It follows easily from Lemma 1.4(4). Corollary 1.7. Let L be a lattice generated by a finite set X which satisfies condition ( ). If Y is a nonempty, proper subset of X, then we have the following covers in L: Y Y (X Y) and Y Y (X Y).

22 2. FREE LATTICES 17 Thus these are covers in every nontrivial relatively free lattice F V (X). In particular, the atoms of FL(X) are the elements (X {x}), for x X. The coatoms are the elements (X {x}), for x X. Throughout this book we will use the notation x = (X {x}) and x = (X {x}) to denote the atoms and coatoms, respectively, of a finitely generated free lattice FL(X). Now we turn to Whitman s condition, which is the crux of the solution of the word problem for free lattices. Theorem 1.8. The free lattice FL(X) satisfies the following condition: (W) If v = v 1 v r u 1 u s = u, then either v i u for some i, or v u j for some j. Proof: Suppose v = v 1 v r u 1 u s = u but that v i u and v u j for all i and all j. If v x u for some x X, then since x is meet prime, v i x u for some i, contrary to our assumption. Let I be the interval u/v and let FL(X)[I] be the lattice obtained by doubling I. By the above remarks, none of the generators is doubled. This implies that X is a subset of FL(X)[I] and so the identity map on X extends to a homomorphism f : FL(X) FL(X)[I]. Since x / I, λ(x) = x, where λ is the epimorphism defined by (1). Hence λ(f(w)) = w for all w FL(X) and this implies f(w) = w if w / I. Thus it follows from (2) and its dual that f(v) = f(v 1 ) f(v r ) = v 1 v r = v,1 u,0 = u 1 u s = f(u 1 ) f(u s ) = f(u), contradicting the fact that v u. The condition (W) is known as Whitman s condition. Notice that it does not refer to the generating set and so it is inherited by sublattices. Also note that Day s doubling is a procedure for correcting (W) failures. As such it has many uses, see [26]. Corollary 1.9. Every sublattice of a free lattice satisfies (W). Every element of a lattice which satisfies (W) is either join or meet irreducible. The next theorem gives a slight variant from [52] of Whitman s condition which is more efficient for computation and also useful in theoretical arguments.

23 18 I. WHITMAN S SOLUTION TO THE WORD PROBLEM Theorem The free lattice FL(X) satisfies the following condition: (W+) If v = v 1 v r x 1 x n u 1 u s y 1 y m = u, where x i and y j X, then either x i = y j for some i and j, or v i u for some i, or v u j for some j. Proof: Suppose we apply (W) to the inequality v u and obtain x i u = u 1 u s y 1 y m. Then either x i y j for some j, or x i u k for some k. The former implies x i = y j and the latter implies v u k. Thus (W+) holds in either case. The other possibilities are handled by similar arguments. Notice that (W+) replaces the test x i u with the test x i {y 1,...,y m }, which is of course much easier. Also notice that in applying (W+) it is permitted that some of the v i s and u j s are in X. Thus, for example, if v = v 1 v r x 1 x n and u = u 1 u s, then v u if and only if v i u for some i or v u j for some j. Theorems 1.5 and 1.8 combine to give a recursive procedure for deciding, for terms s and t, if s FL(X) t FL(X) known as Whitman s solution to the word problem. 1 To test if s t, the algorithm is used twice to check if both s FL(X) t FL(X) and t FL(X) s FL(X) hold. In Chapter XI we will give a presentation of this algorithm more suitable for a computer (rather than for a human) and study its time and space complexity. Theorem If s = s(x 1,...,x n ) and t = t(x 1,...,x n ) are terms and x 1,...,x n X, then the truth of ( ) s FL(X) t FL(X) can be determined by applying the following rules. (1) If s = x i and t = x j, then ( ) holds if and only x i = x j. (2) If s = s 1 s k is a formal join then ( ) holds if and only if s FL(X) i t FL(X) holds for all i. (3) If t = t 1 t k is a formal meet then ( ) holds if and only if s FL(X) t FL(X) i holds for all i. 1 It is interesting that the word problem for free lattices was first shown to be decidable by Thoralf Skolem in 1920 [121], reprinted in [122]. In fact, Skolem showed that the universal first order theory of lattices is decidable; in particular, the word problem for finitely presented lattices is (uniformly) solvable. This result of Skolem seems to have gone unnoticed by lattice theorists. It was found by Stan Burris in his studies of the history of logic. It is also interesting that Skolem s solution to the word problem for finitely presented lattices is polynomial time; see Chapter XI.

24 2. FREE LATTICES 19 (4) If s = x i and t = t 1 t k is a formal join, then ( ) holds if and only if x i t FL(X) j for some j. (5) If s = s 1 s k is a formal meet and t = x i, then ( ) holds if and only if s FL(X) j x i for some j. (6) If s = s 1 s k is a formal meet and t = t 1 t m is a formal join, then ( ) holds if and only if s FL(X) i t FL(X) holds for some i, or s FL(X) t FL(X) j holds for some j. Proof: Conditions (1), (4) and (5) hold by Corollary 1.5, while (2) and (3) are trivial. Theorem 1.8 shows that free lattices satisfy (6). It is easy to see that all possibilities are covered by (1) (6) and that each of these leads to a genuine reduction (except for (1), which gives the answer directly). We are now in a position to give an easy criterion to determine if a subset of a lattice generates a sublattice isomorphic to a free lattice. Corollary Let L be a lattice which satisfies (W) and let X generate L. Then L is isomorphic to FL(X) if and only if the following condition and its dual hold for all x X and all finite subsets Y X: x Y implies x Y. Proof: If x and y X satisfy x y then the condition with Y = {y} implies that x = y and, by Lemma 1.3 and its dual, each x X is join and meet prime. Of course the identity map on X extends to a homomorphism of FL(X) onto L. Since X in L satisfies (1) (6) of Theorem 1.11, if s and t are terms then s L t L if and only if s FL(X) t FL(X) and thus this map must be an isomorphism. Corollary A subset S of a free lattice FL(X) generates a sublattice isomorphic to a free lattice if and only if for all s S and all finite subsets Y S, and the dual condition holds. s Y implies s Y, The next corollary shows that relatively free lattices cannot be embedded into free lattices except in the trivial cases. This contrasts greatly with the situation for other varieties. For example, every free distributive lattice can be embedded into a free modular lattice, see[48] and [55]. Corollary If V is a nontrivial variety of lattices, other than the variety of all lattices, and X is a set with at least three elements, then F V (X) is not a sublattice of any free lattice FL(Y).

25 20 I. WHITMAN S SOLUTION TO THE WORD PROBLEM Proof: If X is a subset of FL(Y) which generates a sublattice isomorphic to F V (X), then by Lemmas 1.2 and 1.13, F V (X) = FL(X). We will see in Theorem 1.28 below that FL(ω) is a sublattice of FL(3). Thus FL(3) (and hence FL(X) if X 3), generates the variety of all lattices. Of course this means that F V (X) = FL(X) is impossible. 3. Canonical Form The canonical form of an element plays an important role in free lattice theory. In this section we show that each element w of a free lattice has a term of least rank representing it, unique up to commutativity. 2 This term is called the canonical form of w. The phrase unique up to commutativity can be made precise by defining equivalent under commutativity to be the equivalence relation, s t, given by recursively applying the following rules. (1) s, t X and s = t. (2) s = s 1 s n andt = t 1 t n andthereisapermutationσ of {1,...,n} such that s i t σ(i) for all i. (3) The dual of (2) holds. Theorem 1.17 below shows that if two terms both represent the same element of FL(X) and both have minimal rank among all such representatives, then they are equivalent under commutativity. We define s t for terms s and t to mean s FL(X) t FL(X). Note that this is only a quasiorder. Later in this section we will derive the semidistributive laws from the existence of the canonical form. We also show that there is a strong connection between the canonical forms and the arithmetic of free lattices. This will allow us to define canonical form in terms of lattice theoretic properties. The following concept is very important in lattice theory, particularly free lattice theory. Let L be a lattice and let A and B be finite subsets of L. We say that A join refines B and we write A B if for each a A there is a b B with a b. The dual notion is called meet refinement and is denoted A B. Note, however, that A B does not imply B A. The next lemma lists the basic properties of join refinement, all of which are straightforward to prove. 2 Usually the canonical form is described as unique up to commutativity and associativity. However, our definition of lattice terms and their ranks, given at the beginning of the previous section, imply that a term of minimal rank can be associated in only one way.

26 3. CANONICAL FORM 21 Lemma The join refinement relation has the following properties. (1) A B implies A B. (2) The relation is a quasiorder on the finite subsets of L. (3) If A B then A B. (4) If A is an antichain, A B, and B A, then A B. (5) If A and B are antichains with A B and B A, then A = B. (6) If A B and B A, then A and B have the same set of maximal elements. We use the term join refinement because if u = A = B and A B, then u = Aisabetter joinrepresentation of uthanu = B in that its elements are further down in the lattice. We will see that in free lattices there is a unique best join representation of each element, i.e., a representation that join refines all other join representations. Lemma Let t = t 1 t n, with n > 1, be a term such that (1) each t i is either in X or formally a meet, (2) if t i = t ij then t ij t for all j. If s = s 1 s m and s t, then {t 1,...t n } {s 1,...,s m }. Proof: For each i we have t i s 1 s m. Applying (W) if t i is formally a meet and using join primality if t i X, we conclude that either t i s j for some j, or t i = t ij and t ij s for some j. However, since s t, the second case would imply t ij t, contrary to assumption (2). Hence in all cases there is a j such that t i s j. Thus {t 1,...,t n } {s 1,...,s m }. Theorem For each w FL(X) there is a term of minimal rank representing w, unique up to commutativity. This term is called the canonical form of w. Proof: Suppose that s and t are both terms of minimal rank that represent the same element w in FL(X). If either s or t is in X, then clearly s = t. Suppose that t = t 1 t n and s = s 1 s m. If some t i is formally a join, we could lower the rank of t by removing the parentheses around t i. Thus each t i is not formally a join. If there is a t i such that t i = t ij and t ij t for some j, then t i t ij t. In this case we could replace t i with t ij in t, producing a shorter term still representing w, which violates the minimality of the term t. Thus t satisfies the hypotheses of Lemma 1.16, whence {t 1,...,t n } {s 1,...,s m }.