Lattices freely generated by an order and preserving certain bounds

Algebra Univers. 67 (2012) 113–120DOI 10.1007/s00012-012-0171-6Published online February 28, 2012© Springer Basel AG 2012 Algebra Universalis

Lattices freely generated by an order and preservingcertain bounds

H. Lakser

Abstract. In 1964, R. A. Dean presented a far-reaching generalization of P.M. Whit-man’s solution of the word problem for free lattices. Dean considered an order Q andsolved the word problem for the lattice freely generated by Q while preserving certain(not necessarily all) existing finite joins and meets in Q. We present a simpler, morenatural proof of Dean’s result.

1. Introduction

Solutions of the word problem for lattices have been discovered or re-

discovered by several authors—first by T. Skolem. Some of these solutions,

like those in S. Burris [1], G. Czedli [3], R. Freese [8], C. Herrmann [13],

and T. Skolem [16] and [17], provide fast algorithms, while some others, due

to R. P. Dilworth [6], R. A. Dean [5], T. Evans [7], J. C. C. McKinsey [15],

and (for free lattices) P. M. Whitman [18], do not; see page 242 of R. Freese,

J. Jezek, and J. B. Nation [8] for further details. However, Whitman’s al-

gorithm and Dean’s algorithm are what Lattice Theory mostly needs. For

example, [10], [11], and G. Czedli and A. Day [4], heavily use Dean’s result.

Also, see G. Gratzer [9] for a discussion of Dean’s result and some of its ap-

plications.

P. M. Whitman [18] solved the word problem for free lattices. Whitman’s

result was generalized by R. P. Dilworth [6] to lattices freely generated by an

order. R. A. Dean [5] considered an order Q and solved the word problem for

the lattice freely generated by Q and preserving certain (not necessarily all)

existing finite joins and meets in Q.

The present work is based on [14], and our aim is to present a new, simpler

and more natural proof of Dean’s solution to the word problem for lattices.

2. The Theorem

Let Q be an order, and let U and L be sets of nonempty finite subsets of Q

such that supX exists in Q for each X ∈ U and inf X exists in Q for each

X ∈ L. If L is a lattice and ϕ : Q → L is isotone and satisfies ϕ(sup X) =

Presented by G. Czedli.Received November 8, 2009; accepted in final form January 28, 2011.2010 Mathematics Subject Classification: Primary: 06B25.Key words and phrases: lattice, freely generated, order, free product, complemented.

114 H. Lakser Algebra Univers.

∨ϕ(X) for each X ∈ U and ϕ(inf X) =

∧ϕ(X) for each X ∈ L, we say that

ϕ is a (U,L)-morphism. If, in addition, ϕ is an order embedding, that is, if

ϕx ≤ ϕy iff x ≤ y for all x, y ∈ Q, then we say that ϕ is a (U,L)-embedding.

A lattice FL(Q;U,L) along with a (U,L)-embedding η : Q → FL(Q;U,L) is

said to be a free lattice generated by Q and preserving the least upper bounds

in U and the greatest lower bounds in L if FL(Q;U,L) is generated as a

lattice by the set η(Q) and the universal mapping property holds: for each

lattice L and (U,L)-morphism ϕ : Q → L, there is a lattice homomorphism

ϕ′ : FL(Q;U,L) → L with ϕ = ϕ′η. By the usual abstract general nonsense,

the pair η, FL(Q;U,L) is unique up to isomorphism.

In order to characterize η and FL(Q;U,L), we introduce the concepts of

U-ideal and L-filter in Q. A subset (possibly empty) I of Q is said to be

a U-ideal if I is a down-set and X ⊆ I implies supX ∈ I for each X ∈ U.

Dually, a subset (again, possibly empty) D of Q is said to be an L-filter if D

is an up-set and X ⊆ D implies inf X ∈ D, for each X ∈ L. Note that for any

x ∈ Q, the principal down-set ↓x generated by x is a U-ideal, and the principal

up-set ↑x is an L-filter. Clearly, the intersection of any family of U-ideals is a

U-ideal, and, dually for L-filters. (The intersection of two nonempty U-ideals,

or of two nonempty L-filters, can be empty; this is why we must permit the

empty set to be an ideal and a filter.) Thus, the set of all U-ideals and the set

of all L-filters are complete lattices.

In order to construct FL(Q;U,L), we work with Term(Q), the set of lattice

terms with operation symbols ∨ and ∧, with all variables being elements of Q.

Many of our definitions and proofs will proceed by recursion on lengthp, the

length of lattice terms p ∈ Term(Q); we define lengthp by setting lengthx = 1

for all x ∈ Q, and length(p ∨ q) = length(p ∧ q) = lengthp + lengthq; that

is, lengthp is the number of variables, counting repetitions, in p.

By recursion on lengthp, we associate with each p ∈ Term(Q) a U-ideal p

of Q, the lower cover of p, and an L-filter p of Q, the upper cover of p:

Definition 1.

(i) If p ∈ Q, then p = ↓p, and p = ↑p.

(ii) p ∨ q = p∨q, the join in the lattice of U-ideals, and p ∨ q = p∧q = p∩q.

(iii) p ∧ q = p ∧ q = p ∩ q, and p ∧ q = p ∨ q, the join in the lattice of

L-filters.

We now define a binary relation ≤ on Term(Q).

Definition 2. Let p, q ∈ Term(Q). Then p ≤ q iff this follows from the

following rules:

(∧W) p = p0 ∧ p1 with p0 ≤ q or p1 ≤ q.

(∨W) p = p0 ∨ p1 with p0 ≤ q and p1 ≤ q.

(W∧) q = q0 ∧ q1 with p ≤ q0 and p ≤ q1.

(W∨) q = q0 ∨ q1 with p ≤ q0 or p ≤ q1.

(CQ) p ∩ q �= ∅.

Lattices freely generated by an order 115

Conditions (∧W), (∨W), (W∧), (W∨) are called the Whitman Conditions,

while (CQ) is the covering condition for Q.

If p ≤ q, then we write q ≥ p. The usual duality for orders and lattices

can be extended in our case where U and L are dual, ∨ and ∧ are dual, and ≤

and ≥ are dual. Then, by Definition 1, lower cover and upper cover are dual,

and, in Definition 2, Conditions (∧W) and (W∨) are dual, Conditions (∨W)

and (W∧) are dual, and Condition (CQ) is self-dual. We can thus extend the

duality principle to statements about Term(Q).

We now show that the binary relation ≤ on Term(Q) is a quasi-ordering,

that is, a reflexive and transitive relation.

Lemma 1. The relation ≤ on Term(Q) is reflexive; that is, for all p ∈

Term(Q), p ≤ p.

Proof. We proceed by induction on the length of p.

If p ∈ Q, then p ∩ p = (↑p) ∩ (↓p) = {p} �= ∅. Thus, by Condition (CQ),

it follows that p ≤ p.

If p = p0 ∨ p1, then, by the induction hypothesis, p0 ≤ p0 and p1 ≤ p1.

So, by Condition (W∨), p0 ≤ p and p1 ≤ p. Then, by Condition (∨W),

p ≤ p.

If p = p0 ∧ p1, the dual argument (in our extended version of duality)

applies. �

In order to show that ≤ is also transitive, we first prove several lemmas.

Lemma 2. Let p ∈ Term(Q). If x ∈ p and y ∈ p, then x ≤ y.

Proof. We proceed by induction on the length of p.

If p ∈ Q, then p = ↓p, and so x ≤ p in Q. Dually, p ≤ y. Thus, x ≤ y.

If p = p0 ∨p1, then p = p0 ∧p1. So, y ∈ p0 and y ∈ p1. By the induction

hypothesis, p0 ⊆ ↓ y and p1 ⊆ ↓ y. Then p = p0 ∨ p1 ⊆ ↓ y, since ↓ y is a

U-ideal. So, x ≤ y.

The dual argument applies if p = p0 ∧ p1. �

Corollary 1. Let p ∈ Term(Q). If y ∈ p, then p ⊆ ↓ y. If x ∈ p, then

p ⊆ ↑x.

Lemma 3. Let p,q ∈ Term(Q). If p ≤ q, then p ⊆ q and, dually, q ⊆ p.

Proof. By duality, we need only establish that p ⊆ q.

We proceed by induction on the sum of the lengths of p and q.

If p ≤ q follows by the covering condition, let x ∈ p ∩ q. Then, clearly,

↓x ⊆ q and, by Corollary 1, p ⊆ ↓x. Then p ⊆ q. If p ≤ q follows by

Condition (∧W), that is, if p = p0 ∧ p1 and, say, p0 ≤ q, then, by the

induction hypothesis, p0 ⊆ q. Then p = p0 ∧ p1 ⊆ p0 ⊆ q. If p ≤ q follows

by Condition (W∨), that is, if q = q0 ∨ q1 and, say p ≤ q0, then, by the

induction hypothesis, p ⊆ q0. Then p ⊆ q0 ⊆ q0 ∨ q1 = q. If p ≤ q follows

by Condition (∨W), that is, p = p0 ∨ p1 with both p0, p1 ≤ q, then, by


the induction hypothesis, p0 ⊆ q and p1 ⊆ q. Then p = p0 ∨ p1 ⊆ q. If

p ≤ q follows by Condition (W∧), the similar argument applies, concluding

the proof. �

We can now establish transitivity.

Lemma 4. The relation ≤ on Term(Q) is transitive; that is, for all p, q,

r ∈ Term(Q), if p ≤ q and q ≤ r, then p ≤ r.

Proof. Let p, q, r ∈ Term(Q), and let p ≤ q and q ≤ r. We prove by

induction on the sum of the lengths of p, q, and r that p ≤ r.

If p ≤ q by the covering condition, that is, if p ∩ q �= ∅, then, since q ⊆ r

by Lemma 3, it follows that p∩r �= ∅. Thus, in this case p ≤ r by the covering

condition. The dual argument applies if q ≤ r by the covering condition.

So, we may assume that p ≤ q and q ≤ r follow by the Whitman conditions.

If p ≤ q by Condition (∧W), that is, if p = p0 ∧p1 with, say p0 ≤ q, then,

by the induction hypothesis, p0 ≤ r, and, so, p ≤ r by Condition (∧W).

The dual argument applies if q ≤ r by Condition (W∨).

If p ≤ q by Condition (∨W), that is, if p = p0 ∨p1 with p0, p1 ≤ q, then,

by the induction hypothesis, p0, p1 ≤ r. Then p ≤ r by Condition (∨W).

The dual argument applies if q ≤ r by Condition (W∧).

There remain only the case that p ≤ q by Condition (W∧) and q ≤ r by

Condition (∧W), and the dual case, p ≤ q by Condition (W∨) and q ≤ r

by Condition (∨W). So, assume that p ≤ q by Condition (W∧), that is that

q = q0 ∧ q1 with p ≤ q0, q1. Assume, furthermore, that q ≤ r by Condition

(∧W), that is, that, say, q0 ≤ r. Then, by the induction hypothesis, p ≤ r.

By duality, the proof that ≤ is transitive is concluded. �

Since, by Lemmas 1 and 4, ≤ is a quasi-ordering, we can define an equiv-

alence relation on Term(Q) by setting p ≡ q iff p ≤ q and q ≤ p. Denoting

the equivalence class of any p by 〈p〉, we thereby get an ordering ≤ on the

quotient set F = Term(Q)/≡ by setting 〈p〉 ≤ 〈q〉 iff p ≤ q.

Lemma 5. The order F = Term(Q)/≡ is a lattice, whereby 〈p〉∨〈q〉 = 〈p∨q〉

and 〈p〉 ∧ 〈q〉 = 〈p ∧ q〉 for each p, q ∈ Term(Q).

Proof. Let p, q ∈ Term(Q). Since ≤ is reflexive, p, q ≤ p ∨ q by Condition

(W∨); that is, 〈p ∨ q〉 is a common upper bound of 〈p〉, 〈q〉. If 〈r〉 is any

common upper bound of 〈p〉 and 〈q〉, then, by Condition (∨W), p ∨ q ≤ r,

that is, 〈p ∨ q〉 ≤ 〈r〉. Thus 〈p ∨ q〉 is the least upper bound of 〈p〉 and 〈q〉.

That 〈p〉 ∧ 〈q〉 = 〈p ∧ q〉 is the dual argument. �

We now define η : Q → F by setting η(x) = 〈x〉 for each x ∈ Q.

Lemma 6. The mapping η : Q → F , whereby η : x �→ 〈x〉, is a (U,L)-

embedding.


Proof. Let x, y ∈ Q. Then x ≤ y in Term(Q) can only follow from the covering

condition. Thus x ≤ y iff x ∩ y �= ∅ iff ↑x ∩ ↓ y �= ∅ iff x ≤ y in Q. Thus η is

an order embedding.

By duality, it suffices to show that η preserves the least upper bounds

of elements of U. Let X = {x1, . . . , xn} be an element of U with x = supX.

We need only show that 〈x〉 ≤ 〈x1〉∨· · ·∨〈xn〉. Set p = (· · · (x1∨x2)∨· · ·∨xn);

clearly, 〈p〉 = 〈x1〉 ∨ · · · ∨ 〈xn〉. Now, p = ↓x1 ∨ · · · ∨ ↓xn and so X ⊆ p,

a U-ideal. Thus x ∈ p, whereby x ≤ p by the covering condition. Thus

〈x〉 ≤ 〈x1〉 ∨ · · · ∨ 〈xn〉, concluding the proof. �

We now show that the pair η, F = Term(Q)/≡ satisfies the universal map-

ping property for FL(Q;U,L).

Lemma 7. Let L be a lattice, let ϕ : Q → L be a (U,L)-morphism, and let

a ∈ L. Then ϕ−1(↓ a) is a U-ideal in Q and ϕ−1(↑ a) is a L-filter in Q.

Proof. We need only show that ϕ−1(↓ a) is a U-ideal, and appeal to duality.

That ϕ−1(↓ a) is a down-set follows from the fact that ϕ is isotone. That

ϕ−1(↓ a) is then a U-ideal follows from the fact that ϕ preserves joins of ele-

ment of U; if X ∈ U and X ⊆ ϕ−1(↓ a), then ϕ(X) ⊆ ↓ a and so ϕ(∨

X) =∨

ϕ(X) ≤ a. �

We now proceed to establish the universal mapping property. Let L be a

lattice and let ϕ : Q → L be a (U,L)-morphism. Then ϕ extends to a mapping

f : Term(Q) → L with f(p ∨ q) = f(p) ∨ f(q) and f(p ∧ q) = f(p) ∧ f(q),

by substituting ϕ(x) for each variable x. To get our desired homomorphism

ϕ′ : F → L, we need only show that f(p) = f(q) whenever p ≡ q, for which

it suffices to show that f(p) ≤ f(q) whenever p ≤ q.

Lemma 8. Let p ∈ Term(Q). If x ∈ p, then ϕ(x) ≤ f(p), and if x ∈ p, then

ϕ(x) ≥ f(p).

Proof. By duality, we need only show that ϕ(x) ≤ f(p) if x ∈ p, that is, that

p ⊆ ϕ−1(↓ f(p)).

We proceed by induction on the length of p.

If p ∈ Q, then, clearly, p = ↓p ⊆ ϕ−1(↓ϕ(p)) = ϕ−1(↓ f(p)).

If p = p0 ∧ p1, then, by the induction hypothesis, p0 ⊆ ϕ−1(↓ f(p0)) and

p1 ⊆ ϕ−1(↓ f(p1)). Therefore,

p = p0 ∩ p1 ⊆ ϕ−1(↓ f(p0)) ∩ ϕ−1(↓ f(p1)) = ϕ−1(↓ f(p0) ∩ ↓ f(p1))

= ϕ−1(↓ (f(p0) ∧ f(p1))) = ϕ−1(↓ f(p)).

If p = p0 ∨ p1, then again by the induction hypothesis, p0 ⊆ ϕ−1(↓ f(p0))

and p1 ⊆ ϕ−1(↓ f(p1)). Since f(p0), f(p1) ≤ f(p0) ∨ f(p1) = f(p), we con-

clude that p0, p1 ⊆ ϕ−1(↓ f(p)). Since ϕ−1(↓ f(p)) is a U-ideal by Lemma 7,

it follows that p = p0 ∨ p1 ⊆ ϕ−1(↓ f(p)), concluding the proof. �


We can now easily show that f factors through ≡.

Lemma 9. If p, q ∈ Term(Q) and p ≤ q, then f(p) ≤ f(q).

Proof. We proceed by induction on the sum of the lengths of p and q.

If p ≤ q by the covering condition, let x ∈ p ∩ q. Then, by Lemma 8,

f(p) ≤ ϕ(x) ≤ f(q).

Otherwise, p ≤ q follows from one of the Whitman conditions, and, using

the induction hypothesis, the derivation is straight-forward. For example, if

p ≤ q by Condition (∧W), that is, if p = p0 ∧ p1 with, say p0 ≤ q, then, by

the induction hypothesis, f(p0) ≤ f(q), and so

f(p) = f(p0) ∧ f(p1) ≤ f(p0) ≤ f(q).

If p ≤ q by Condition (∨W), that is, if p = p0 ∨ p1 with p0, p1 ≤ q, then

f(p0), f(p1) ≤ f(q) and so f(p) = f(p0) ∨ f(p1) ≤ f(q). The other two

Whitman conditions are the dual of these. �

We thus have our main result, Dean’s Theorem.

Theorem. The lattice F = Term(Q)/≡, where, for p, q ∈ Term(Q), the

relation p ≤ q is as in Definition 2 and where p ≡ q iff p ≤ q and q ≤ p,

along with the mapping η : Q → F with η : x �→ 〈x〉, the ≡ equivalence class of

x, is the free lattice FL(Q;U,L).

Proof. By Lemma 5, F is a lattice. By Lemma 6, the mapping η : Q → F is a

(U,L)-embedding. Clearly, F is generated by η(Q).

Let L be a lattice, and let ϕ : Q → L be a (U,L)-morphism. By the para-

graph following the proof of Lemma 7, we have f : Term(Q) → L with f(x) =

ϕ(x) for all x ∈ Q, and with f(p∨q) = f(p)∨f(q) and f(p∧q) = f(p)∧f(q)

for all p, q. By Lemma 9, ϕ′ : F → L, whereby ϕ′ : 〈p〉 �→ f(p), is well defined,

and so is the desired lattice homomorphism ϕ′ with ϕ = ϕ′η, establishing the

universal mapping property �

We close by relating the lower cover p and the upper cover p to the lattice

structure of FL(Q;U,L).

Lemma 10. For each p ∈ Term(Q), p = {x ∈ Q | η(x) ≤ 〈p〉 } and, dually,

p = {x ∈ Q | η(x) ≥ 〈p〉 }.

Proof. By duality, we need only show the first assertion. If x ∈ p, then

x ∈ ↑x ∩ p = x ∩ p. Then, by the covering condition, x ≤ p, that is,

η(x) = 〈x〉 ≤ 〈p〉. (Alternatively, that η(x) ≤ 〈p〉 is just Lemma 8 with

L = FL(Q;U,L) and ϕ = η.) On the other hand, if x ≤ p, then, by Lemma 3,

↓x = x ⊆ p and so x ∈ p. �

Thus, if we identify Q with the (isomorphic) suborder η(Q) of FL(Q;U.L),

then p = ↓ 〈p〉 ∩ Q and p = ↑ 〈p〉 ∩ Q.

Finally, we remark that the sets in U and L can be of arbitrary finite size.

However, we do not know of any applications where these sets are other than


pairs. We present, though, a trivial example where these sets are triples. Let

Q be the order depicted in Figure 1, and let U = L = {{a, b, c}}. Then

FL(Q;U,L) is the free lattice generated by {a, b, c}, while FL(Q; ∅, ∅) is the

free lattice with a new zero, 0, and a new unit, 1, adjoined.

Figure 1

3. Dean’s approach

In [5], Dean’s approach was not via covers. Rather, inspired by the approach

in R. P. Dilworth [6], he gave a multi-step definition of x ≤ p and p ≤ x for

each x ∈ Q and p ∈ Term(Q). In relation to our approach, these definitions

amount to algorithms for deciding x ∈ I ∨J , where I and J are either U-ideals

or L-filters. Rather than Condition (CQ) he has

p ≤ x and x ≤ q for some x ∈ Q.

Later, he sets p = {x ∈ Q | x ≤ 〈p〉 } and p = {x ∈ Q | x ≥ 〈p〉 } to get

Condition (CQ).

The use of upper and lower covers to define a lattice generated “freely”,

in some sense, by an order seems to have originated with C. C. Chen and

G. Gratzer [2], and then used in G. Gratzer, H. Lakser, and C. R. Platt [12];

in both of these, the covers were elements of the underlying order.

In H. Lakser [14], the covers were generalized to be U-ideals and L-filters,

giving the approach in Section 2 to Dean’s Theorem.

Interestingly, the first use of covers is the far earlier paper of R. P. Dil-

worth [6] mentioned above; however, they do not serve the same purpose as in

this paper.

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H. Lakser

Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canadae-mail : [email protected]

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Lattices freely generated by an order and preserving certain bounds