COPRODUCTS OF BOUNDED

DISTRIBUTIVE LATTICES: CANCELLATION

A Banff Conference Problem

Jonathan David Farley

Abstract. Let L∗M denote the coproduct of the bounded distributive lattices L and M. At the 1981 Banff Conference on Ordered Sets, the following question was posed: What is the largest class L of finite distributive lattices such that, for every non-trivial Boolean lattice B and every L ∈ L, B ∗ L = B ∗ L′ implies L = L′?

In this note, the problem is solved.

1. Introduction.

A Post algebra is a bounded distributive lattice (with least element 0 and greatest element 1) with a non-trivial finite chain 0 = c_{0} < ⋯ < c_{n−1} = 1 and a Boolean sublattice B such that every element has a unique representation

⋁_{i=1}^{n−1}(b_{i} ∧ c_{i}) where b_{1} ⩾ … ⩾ b_{n−1} are in B. The chain is called the chain of

constants. It is well known that Post algebras are exactly the coproducts B ∗ C of non-trivial Boolean lattices B and non-trivial finite chains C in the category of bounded distributive lattices ([13], Theorem 2). It is also well known that the chain of constants in a Post algebra P is unique in the following sense: If B is a Boolean sublattice of P and C and C′ are finite subchains containing 0 and 1 such that P = B ∗ C = B ∗ C′, then C = C′. That is, C and C′ are equal as subsets, not simply isomorphic ([7], Theorem 2.1). More precisely, if ι_{B} : B ↪ B ∗ C, ι_{C} : C ↪ B ∗ C, ι′_{B} : B ↪ B ∗ C′, and ι_{C′} : C′ ↪ B ∗ C′ are

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 06D25, 54F05, 06E15, 54A05, 54B10, 54D05.

Key words and phrases. (generalized) Post algebra, distributive lattice, Boolean lattice, coproduct, Priestley duality, (partially) ordered topological space.

The author would like to thank Dr. H. A. Priestley for comments regarding the presentation of this paper. He would also like to thank Dr. Stephen Comer for bringing to his attention reference [2].

the natural monomorphisms and Ψ : B ∗ C ≅ B ∗ C′ an isomorphism, then the image of Ψ ∘ ι_{C} is the image of ι_{C′}.

Balbes and Dwinger proved that if B and B′ are non-trivial Boolean lattices and C and C′ non-trivial bounded chains, then B ∗ C = B′ ∗ C′ implies B = B′ and C ≅ C′ ([1], Theorem 1). Moreover, for a given non-trivial bounded chain C, B ∗ C = B ∗ C′ implies C = C′ for all non-trivial Boolean lattices B and all bounded chains C′ if and only if C is rigid, that is, has just one automorphism ([1], Theorem 6). Comer and Dwinger investigated bounded distributive lattices K such that, for all bounded distributive lattices L and M in a given class, K ∗ L = K ∗ M implies L = M ([3]).

At the 1981 Banff Conference on Ordered Sets, F. Yaqub asked for a description of the largest class L of finite distributive lattices such that, for every non-trivial Boolean lattice B and every L ∈ L, B ∗ L = B ∗ L′ implies L = L′ ([12], p. 849).

The problem as stated must be formulated more precisely, but we answer the question, as well as the corresponding one for isomorphisms (Theorems 2 and 3). Our tool is Priestley duality for distributive lattices.

2. Definitions, Notation, and Basic Results.

For basic notions, see [6]. A component of a poset is a non-empty maximal connected subset. A subset U of a poset P is an up-set if, for all u ∈ U and p ∈ P, u ⩽ p implies p ∈ U. A partially ordered topological space P is totally order-disconnected if, for all p, q ∈ P such that p ⩽̸ q, there exists a clopen up-set U such that p ∈ U and q ∉ U. A Priestley space is a compact totally order-disconnected space.

Given an ordered space P, let D(P) denote the bounded distributive lattice of clopen up-sets. Given a bounded distributive lattice L, let P (L) denote the Priestley space of prime filters, with topology generated by the subbasis

{{F ∈ P (L) | a ∈ F }, {F ∈ P (L) | a ∉ F } | a ∈ L}.

The operators D and P extend to functors which yield a dual equivalence between the categories of bounded distributive lattices with {0, 1}-preserving homomorphisms and Priestley spaces with continuous order-preserving maps. For some consequences of Priestley duality, see [9] and [10].

COPRODUCTS OF BOUNDED DISTRIBUTIVE LATTICES: CANCELLATION 3

It is well known that P (L) is an antichain if L is a Boolean lattice ([6], Theorem 9.8). Also, P (L) is a finite poset with the discrete topology if L is a finite distributive lattice. Further, P (L × M) is order-homeomorphic (orderisomorphic via a map that is a homeomorphism) to P (L) + P (M), the disjoint sum of the ordered spaces ([6], Exercise 10.3(iii)). Lastly, P (L ∗ M) is orderhomeomorphic to P (L)×P (M). Hence, for bounded distributive lattices L and M, L ∗ M is isomorphic to L^{P (M)}, the lattice of continuous order-preserving maps from P (M) to L with the discrete topology, ordered pointwise; L is associated with the constant maps, and M with the maps 2^{P (M)}, where 2 = {0, 1} ([5], Theorem and Corollary, [4], Corollary 2.3, and [11], Theorem). In [14] and [15], this aspect of duality is used to understand generalized Post algebras, coproducts of Boolean lattices and bounded distributive lattices. (See Figures 1–4.)

[figure]

Figure 1. The poset P and the lattice L = D(P).

[figure]

Figure 2. The poset Q and the lattice M = D(Q).

[figure]

Figure 3. The poset P × Q.

3. The Solution to the Problem for Isomorphisms and Equality. We must emend Yaqub’s problem because of the following example: Let B

be a Boolean lattice such that B^{2} ≅ B. (For instance, let B := 2^{ω}.) Then for any finite poset P, B^{P} ≅ B^{(2×P)}. Hence, for any finite distributive lattice L, B ∗ L ≅ B ∗ L^{2}. Therefore, in the problem, we must restrict the class to which L′ can belong. We insist that L and L′ belong to the same class.

A class L of finite distributive lattices is shabazz if, for all lattices M and N, M × N^{2} ∈ L implies M × N ∈ L. A finite distributive lattice is square-free if it has no direct factor of the form N^{2} (N a non-trivial lattice). Let B be the class of non-trivial Boolean lattices.

[figure]

Figure 4. The lattice L ∗ M ≅ D(P × Q).

Lemma 1. Let X be an antichain, P and P′ posets, C a component of P, and Ψ : X × P ≅ X × P′ an order-isomorphism. Let π_{X} : X × P′ → X be the projection.

Then, for all x ∈ X, there exists x′ ∈ X such that

(π_{X} ∘ Ψ)[{x} × C] = {x′}.

Proof. Suppose that Ψ(x, p_{0}) = (x′, p′_{0}) and Ψ(x, p_{1}) = (x′′, p′_{1}). If p_{0} ⩽ p_{1}, then (x′, p′_{0}) ⩽ (x′′, p′_{1}), so that x′ = x′′. □

Theorem 2. Let L be a shabazz class of finite distributive lattices. The following are equivalent:

(1) for all B ∈ B and L, L′ ∈ L, B ∗ L ≅ B ∗ L′ implies L ≅ L′;

(2) every member of L is square-free.

Proof. Assume L ≅ M × N^{2} for some L ∈ L and lattices M and N. Let L′ := M × N and let B := 2^{ω}. As B ≅ B^{2}, we have B ∗ L ≅ B ∗ L′.

Now assume B ∈ B and let L and L′ be square-free finite distributive lattices. Then B ∗ L ≅ B ∗ L′ implies there exists an order-isomorphism Ψ : X × P ≅ X × P′ where X := P (B), P := P (L), and P′ := P (L′); P and P′ are finite posets with pairwise non-isomorphic components. Let π_{P′} : X × P′ → P′ be the projection and fix x ∈ X. By Lemma 1, the map

p ↦ (π_{P′} ∘ Ψ)(x, p) (p ∈ C)

is an order-isomorphism on each component C of P. □

Theorem 3. Let L be a class of finite distributive lattices. The following are equivalent:

(1) for all B ∈ B and L, L′ ∈ L, B ∗ L = B ∗ L′ implies L = L′;

(2) every member of L is rigid.

Proof. Let L be a finite distributive lattice and ϕ : L ≅ L a non-trivial automorphism. Define Φ : L^{2} → L^{2} by Φ(a, b) = (a, ϕ(b)) (a, b ∈ L). Then Φ is an automorphism of 2^{2} ∗ L that does not map constants to constants.

Now let X be a non-empty Priestley space that is an antichain. Let P and P′ be finite rigid posets. Let Ψ : X × P ≅ X × P′ be an order-homeomorphism. We must show that, for all U ∈ D(P), there exists U′ ∈ D(P′) such that Ψ[X × U] = X × U′.

Without loss of generality, P and P′ are connected. By Lemma 1, for all x ∈ X, there exists x′ ∈ X such that Ψ[{x} × P] = {x′} × P′. For all x ∈ X and U ∈ D(P), let U′_{x} ∈ D(P′) be such that Ψ[{x} × U] = {x′} × U′_{x}.

It suffices to show that, for all x_{0}, x_{1} ∈ X and p ∈ P,

(π_{P′} ∘ Ψ)(x_{0}, p) = (π_{P′} ∘ Ψ)(x_{1}, p).

By Lemma 1, the maps p ↦ (π_{P′} ∘ Ψ)(x_{0}, p) and p ↦ (π_{P′} ∘ Ψ)(x_{1}, p) (p ∈ P) are order-isomorphisms, so, by rigidity, they are equal. □

4. Related Results.

In [3], a research announcement of results that apparently were never published*, it is stated that if K, L, and M are bounded distributive lattices with L and M rigid, then K ∗ L = K ∗ M implies L = M. (The author discovered Theorem 3 independently of [3].) The interpretation of this statement is not the same as in our paper, as is evidenced by the case where K, L, and M are each the three-element chain (cf. [8], Fig. 4).

In [2], Theorem 2.4(i), it is shown that there exist countable Boolean algebras L and L′ such that 2^{2} ∗ L ≅ 2^{2} ∗ L′ but L ≇ L′. It also follows from (1.4) in [2] that, for non-trivial finite distributive lattices L, L′, and M, L ∗ M ≅ L′ ∗ M implies L ≅ L′.

References

1. R. Balbes and Ph. Dwinger, Uniqueness of representations of a distributive lattice as a free product of a Boolean algebra and a chain, Colloquium Mathematicum 24 (1971), 27–35.

*The author was unsuccessful in his attempt to obtain a copy of the (unpublished) paper announced in [3].

2. Stephen D. Comer, Arithmetic properties of relatively free products, Proceedings of the University of Houston Lattice Theory Conference, Siemion Fajtlowicz and Klaus Kaiser (eds.), Department of Mathematics, University of Houston, Houston, Texas, 1973, pp. 180–193.

3. Stephen D. Comer and Philip Dwinger, Cancellation for bounded distributive lattices: preliminary report, Notices of the American Mathematical Society 21 (1974), A-44.

4. William H. Cornish, Ordered topological spaces and the coproduct of bounded distributive lattices, Colloquium Mathematicum 36 (1976), 27–35.

5. Brian A. Davey, Free products of bounded distributive lattices, Algebra Universalis 4 (1974), 106–107.

6. B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 1990.

7. Ph. Dwinger, Notes on Post algebras. I, Indagationes Mathematicae 28 (1966), 462–468.

8. Bjarni Jónsson, Arithmetic of ordered sets, Ordered Sets, Ivan Rival (ed.), D. Reidel, Dordrecht, Holland, 1982, pp. 3–41.

9. H. A. Priestley, Ordered topological spaces and the representation of distributive lattices, Proceedings of the London Mathematical Society 24 (1972), 507–530.

10. H. A. Priestley, Ordered sets and duality for distributive lattices, Annals of Discrete Mathematics 23 (1984), North-Holland, Amsterdam, 39–60.

11. R. W. Quackenbush, Free products of bounded distributive lattices, Algebra Universalis 2 (1972), 393–394.

12. Ivan Rival (ed.), Ordered Sets, D. Reidel, Dordrecht, Holland, 1982.

13. G. Rousseau, Post algebras and pseudo-Post algebras, Fundamenta Mathematicae 67 (1970), 133–145.

14. T. P. Speed, A note on Post algebras, Colloquium Mathematicum 24 (1971), 37–44.

15. Fawzi M. Yaqub, Representation of Post L-algebras by rings of sets, Rendiconti dell’ Istituto di Matematica dell’ Università di Trieste 14 (1982), 32–40.

Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240, United States of America

Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, California 94720, United States of America

E-mail address: farley@math.vanderbilt.edu