Whitman's Solution to the Word Problem

http://dx.doi.org/10.1090/surv/042/01

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 We prove an unpublished result of the first author that there is a unary polynomial function on the free lattice with three generators without a fixed point.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 Jonsson 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 Gratzer [73]. For the most part we follow the notation and terminology of [100]. We use the notation A C B for set inclusion, while A C B denotes proper set inclusion. An order relation on a set P is a binary relation which satisfies (1) x < x, for all x G P (reflexivity) (2) x < y and y < x imply x = y, for all x, y G P (anti-symmetry) (3) x < y and y < z imply x < z, for all x, y, z G P (transitivity) An ordered set (also known as a partially ordered set) is a pair P = (P, y if and only if y < x. (P, >) is an ordered set known as the dual of (P, 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

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10

I. WHITMAN'S SOLUTION TO THE WORD PROBLEM

dual. Thus L[C] is a lattice and it follows from equation (2) and its dual that A is a homomorphism which is clearly onto L. • 2. Free Lattices Since the lattice operations are both associative, we define lattice terms over a set X, and their associated lengths, in a manner 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\,..., tn are terms of lengths k\,..., /cn, then (£i V • • • V tn) and (t\ A • • • A tn) are terms with length 1 + k\ + • • • + kn. When we write a term we usually omit the outermost parentheses. Notice that if x, y, and z £ X then xV yV z

x\t (y\f z)

(x\/ y)\/ z

are all terms (which always represent the same element when interpreted into any lattice) but the length of x V y V 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 lit £ X, and if t = t\ V • • • V tn or t = t\ A • • • A tn, where n > 1, then the complexity of t is one more than the maximum of the complexities of ti,..., tn. 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\ V • • • \ltn or t = t\ A • • • A£ n , then the subterms of t consist of t together with the subterms of £ i , . . . , tn. By the phrase ' £ ( # i , . . . , xn) is a term' we mean that t is a term and xi,...,xn are (pairwise) distinct variables including all variables occurring in t. If t(x\,..., xn) is a term and L is a lattice, then £L denotes the interpretation of t in L, i.e., the induced n-ary operation on L. If a i , . . . , a n G L, we will usually abbreviate t L ( a i , . . . , an) by £ ( a i , . . . , an). Very often in the study of free lattices, we will be considering a lattice L with a specific generating set {x\,... ,xn}. In this case we will use t L to denote t L ( ^ i , . . . , xn). If s(xi,..., xn) and t(x\,..., xn) 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 C F. We say that F is freely generated by X if X generates F and every map from X into any lattice L extends to a lattice homomorphism of F into L. Since X generates F, such an extension is unique. It follows easily that if F i is freely generated by X\ and F 2 is freely generated by X2 and |Xi| = |X 2 |, then F i and F 2 are isomorphic. Thus, if X is a set, a lattice freely generated by X is unique up to isomorphism. We will see that such a lattice always exists. 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,


2. F R E E L A T T I C E S

11

provided we identify each element of x G 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 G 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 G T(X) represents a G L if tL =a. As Jonsson 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 Fy(X) and refer to it as the relatively free lattice in V over X. If £ is the variety of all lattices, then, in this notation, F,c (X) — FL(X). However, because of tradition, we will use FL(X) to denote the free lattice. The relatively free lattice Fy(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. LEMMA 1.2. Let V be a nontrivial variety of lattices and let Fy(X) relatively free lattice in V over X. Then (t)

/\S

/\ 5, for some finite S C Y. But every element of Y is above a; hence a = /\ 5, as desired. Of course (2) is proved dually. Let T be a finite, nonempty subset of X and let a — f\T. It follows from condition (f) 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 = FU I. If F fl I is nonempty, there would be subsets S and T of X with /\S which is of course much easier. Also notice that in applying (W+) it is permitted that some of the v^s and u^-'s are in X. Thus, for example, if v = v\ A • • • A vr A



14

X\ A • • • A xn and u = u\ V • • • V u s , then v < u if and only if Vi < u for some i or v < Uj for some j . Theorems 1.5 and 1.8 combine to give a recursive procedure for deciding, for terms s and £, if s F L ( x ) < £ F L W known as Whitman's solution to the word problem. 1 To test if s ~ £, the algorithm is used twice to check if both s F L ( x ) < ^FL(X) a n d ^FL(X) < S F L ( X ) n o l d j n 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 1.11. If s = s(xi,... xi,...,xn G l , then the truth of (*)

S

FL

,xn)

and t = t(x\,...

,xn)

are terms and

W < *FLW

can be determined by applying the following rules. (1) If s = Xi and t = Xj, then (*) holds if and only Xi = Xj. (2) / / s = si V • • • V Sk is a formal join then (*) holds if and only if si ^ ' < t F L W holds for alii. (3) If t = ti A " • A tk is a formal meet then (*) holds if and only if s F L ( x ) < ti ^ ' holds for all i. (4) If s = xi and £ = £i V • • • V £& is a formal join, then (*) holds if and only if ^

,FL(X) r

Xi < tj for some j . (5) If s = s\ A - — A Sk is a formal meet and t = xi, then (*) holds if and only . , FL(X) ^

/•

y Sj < xi for some j . (6) If s = S\A" -Ask is a formal meet and t = t\ V • • • V£m is a formal join, then (*) holds if and only if sJL{x) 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. LEMMA

1.15. The join refinement relation has the following properties.

(1) A < B implies \J A 1, and let a = V ^ i ai- Clearly m

w =

vVaVan=v\/a\/\Jbj, 3=1

and it follows by P(l,ra) that 771

w = v\/ a V \f (an Abj). 3=1

Therefore 71—1

771

771

771

w = (*> V V (a n A 6,)) V \ / ^ = (v V \J {an A bj)) V \J bj, 3=1

i=l

3=1

3=1

and it follows by P(n — 1, m) that 77i

n —1

77i

n

m

w = v V \ / (a n A 6,-) V \ / Y (a< A 6j) = « V V V ( a i jf=l

i=l j=l

i=l

j=l

A 6

^'

a 4. Continuity A subset A of a lattice L is said to be wp directed if every finite subset of A has an upper bound in A. It suffices to check this for pairs: A is up directed if for all a, b € A there exists c G A with a < c and 6 < c. Clearly, any join closed set is up directed. A down directed set is defined dually. A lattice is said to be upper continuous if whenever A is an up directed set having a least upper bound u = \J A, then, for any 6, (4)

\J (a A b) = u A b. aEA

Lower continuous is defined dually. A lattice is continuous if it is both lower and upper continuous. Often it is assumed that continuous lattices are complete, but we do not make that assumption here as free lattices are not complete; see Example 1.24.3 Now let us prove Whitman's result that free lattices are continuous. 3 In the standard definition of upper and lower continuity for a complete lattice, the set A is assumed to be a chain (rather than a directed set). It is then shown that continuity for chains implies continuity for directed sets. However, this need not be true for lattices which are not complete. Since the stronger condition for directed sets is true in free lattices, and is in fact needed for some arguments, it makes sense to take the directed set version as the definition of continuity for lattices which are not necessarily complete.



20

Free lattices are continuous.4 show that free lattices are lower continuous; the result then follows by duality. Suppose that u = f\A for a down directed set, and let b be arbitrary. Clearly u V b is a lower bound for {a V b : a G A}. Suppose that it is not the greatest lower bound and let c be an element of minimal rank such that T H E O R E M 1.22. P R O O F . We will

(5)

c^uVb,

c < a V b for all a G A.

If c < a for all a G A, then c • • •»x n G {£1, xn}, then ([xn V £1) A £2) • • • V xn is a fixed point. The proof will be omitted.

6. Sublattices of Free Lattices In this section we prove Whitman's theorem that FL(CJ) is a sublattice of FL(3), and Jonsson and Kiefer's theorem that there is a nontrivial equation satisfied by all finite sublattices of a free lattice. We begin with the following result of Galvin and Jonsson [65], which applies to all varieties of lattices. In fact it even applies to a wider class of algebras which includes Boolean algebras.


6. SUBLATTICES OF FREE LATTICES

23

T H E O R E M 1.27. Every relatively free lattice ¥y(X), and hence every sublattice of a relatively free lattice, is a countable union of antichains. Thus relatively free lattices, and sublattices of relatively free lattices, contain no uncountable chains. P R O O F . The result is obvious if X is finite, so assume X is infinite and that XQ is a countable subset of X. Let G be the group of automorphisms of Fv(-X") which are induced from the permutations of X fixing all but finitely many x € X. For u and v G Fy(X), let u ~ G V denote the fact that u and v lie in the same orbit, i.e., there is a a G G such that a(u) = v. Notice that u ~ G V means that u and v can be represented by the same term except that the variables are changed. For every element u G Fy(X), there is a v in the sublattice generated by XQ with u ~ G v- Thus F v P O has only countably many orbits under G. (An orbit is just an equivalence class of ~ G - ) Let a G G and suppose u < a(u). Then by applying a repeatedly to this inequality, we obtain

(9)

u
n. Since / 5 m (a;Vy) = dmWy < 4 V | / = /5 n (xVy), ra > n. Hence m = n and thus the iu n 's form an antichain. Now suppose that w m < w n i V- • -Vwnk for some distinct, positive m, n i , . . . , n^. Then c m A (d m V y) < iu ni V • • • V it;nfc and we apply (W). By Lemma 1.29, neither x nor y lie below wni V • • • V u>nfc, and it follows that the only possibility is c m A (dmWy) < wni, for some i. But, by joining both sides with z, this gives wm < wn., a contradiction. Now suppose wni A • • • A wUk < wm. No wn < z and hence wni A • • • A wnfc ^ 2. This fact, and the incomparability of the w n 's, imply that zn, n = 1,2,..., is isomorphic to FL(u;). • Next we present Jonsson and Kiefer's theorem. T H E O R E M 1.30. Let L be a lattice satisfying (W). Suppose elements a\, a2, as, and v E L satisfy (1) Q>i ^ %' V a^ V v whenever {i,j,k} = {1,2,3}, (2) v£aifori = l, 2, 3, (3) v is meet irreducible. Then L contains a sublattice isomorphic to FL(3). P R O O F . For {ij,k} = {1,2,3}, let b{ = a{ V [(a3Wv) A (ak Vv)]. Iffy < ^ V6fc then ai < bi < bj V 5^ < a7 V a^ V v, contradicting (1). Thus the fy's are join irredundant. In particular they are pairwise incomparable. Suppose b\ A 62 < &3 = 03 V[(ai W)A(a2Vv)], and apply (W). Neither b\ nor 62 is below 63 and if 61A62 < a3, then v < 03, contradicting (2). Hence 61 A 62 < (ai V v) A (a2 V f) < a\ V i>, and we apply (W) again. Since v < b\ A 62, the inequality 61 A 62 < ^i would imply i; < a\ and so cannot occur. If b\ A 62 < v, then i; = 61 A 62 and so would be a proper meet, which contradicts (3). If 62 < a\ V v, then a2 < ai V v, which violates (1). Thus we must have b\ < a\ V v which implies

(c&2 V v) A (03 V v ) < a i V v . By (1) neither meetand is contained in a\ Vv and by (2) (0,2 Vv) A (a3 Vu) ^ 01. The last possibility gives v = («2 V v) A (03 V v), contradicting (3). Hence 61 A 62 ^ 63, and symmetrically fy A fy ^ 6/-. Thus, by Corollary 1.13, the sublattice generated by 61, 62, and 63 is isomorphic to FL(3). • A lattice L is said to have breadth at most n if whenever a G L and 5 is a finite subset of L such that a = \J 5, there is a subset T of 5, with a = \jT and |T| < n. The breadth of a lattice is the least n such that it has breadth at most n. The reader can verify that this concept is self dual. COROLLARY 1.31. / / L is a finite lattice satisfying (W), then the breadth of L is at most 4. The variety generated by finite lattices which satisfy (W) is not the variety of all lattices. In particular, finite sublattices of a free lattice have breadth at most four and satisfy a nontrivial lattice equation.


6. S U B L A T T I C E S O F F R E E L A T T I C E S

25

P R O O F . Suppose L is a finite lattice satisfying (W) and a = a\ V • • • V an holds for some a G L and some n > 4 and that this join is irredundant. Then, since every element of a lattice which satisfies (W) must be either join or meet irreducible, v —