Boolean products and canonical extensions of bounded distributive lattice expansions
by
Bjarni Jonsson
Vanderbilt University

The talk will be based on a joint paper with Mai Gehrke, Bounded distributive lattice expansions, that is to appear in Mathematica Scandinavica. The paper, still under revision, is available on the authors' home pages. In this abstract, the construction of canonical extensions will be described in some detail, but little space will be given to their study. In the talk itself, this will be reversed; the construction will be dealt with very sketchily, but certain techniques used in investigating canonical extensions will be the principal topic.

A bounded distributive lattice expansion A=(A₀, \omega^A, \omega in O) is a bounded distributive lattice A₀=(A, \/ , /\ , 0, 1) with auxiliary operations \omega^A. Canonical extensions of bounded distributive lattice expansions have been defined and investigated under some conditions on the auxiliary operations, but here these operations will be completely arbitrary.

It is convenient to have names for the several categories involved in the definition:

• DL: The category of all bounded distributive lattices and their homomorphisms.
• PS: The category of all Priestley spaces, with continuous isotone maps as the morphisms
• PO: The category of all partially ordered sets (posets), with isotone maps as the morphisms.
• DL⁺:The category of all doubly algebraic, bounded distributive lattices, with complete homomorphisms as the morphisms.
• DLE_\mu: The category of all bounded distributive lattice expansions of similarity type \mu, and their homomorphisms. (When the similarity type is clear from the context, or is irrelevant, the subscript may be omitted.)

The name of a category will also be used as a collective name for its objects, E. g., a bounded distributive lattice will be referred to as a DL.

As in the earlier work, canonical extension will be defined in three stages. For a DL A, the canonical extension A^\sigma is defined, as before, using Priestley duality between DL and PS and the duality between DL⁺ and PO. The Priestley dual of a DL A is a PS A_\delta=(A_\delta, \tau, <= ), whose dual (A_\delta)^\delta, also referred to as the second dual of A, is a DL that is isomorphic to A. On the other hand, if F is the functor that forgets the topology, then F(A_\delta)=(A_\delta, <= ) is a poset whose dual F(A_\delta) ^* is a DL⁺ containing (A_\delta)^\delta as a sublattice. Combined with the natural isomorphism from A onto its second dual, this yields an injective homomorphism from A into the DL⁺ (A_\delta) ^* . Since we prefer to work with extensions rather than injective homomorphisms, we define the canonical extension A^\sigma of a DL A up to equivalence to be a DL⁺ containing A as a sublattice, such that the natural isomorphism A\backsimeq (A_\delta)^\delta extends to an isomorphism A^\sigma\backsimeq (A_\delta) ^* . Denoting by J^\infty(A^\sigma) and by M^\infty(A^\sigma) the sets consisting, respectively, of all the strictly join irreducible elements of A^\sigma and of all the strictly meet irreducible elements of A^\sigma, we can characterize A^\sigma abstractly by the condition that it is a DL⁺ containing A as a sublattice, with the following two properties:

(Sep) For all p in J ^\infty(A^\sigma) and u in M^\infty(A^\sigma), if p <= u, then there exists a in A with p <= a <= u.

(Comp) For all X, Y subset or equal A, if /\ X <= \/ Y, then there exist finite sets F subset or equal X and G subset or equal Y with /\ F <= \/ G.

An element x in A^\sigma is said to be open (x in O(A^\sigma)) if x = \/ X for some X subset or equal A, and x is said to be closed (x in K(A^\sigma) if x= /\ X for some X subset or equal A This terminology is suggested by the isomorphism between A^\sigma and (A_\delta) ^* . The elements of (A_\delta) ^* can be thought of as order filters in the Priestley space A_\delta, and the open and the closed elements of A^\sigma are precisely the elements that correspond to open and to closed order filters, respective.

We next define the canonical extension f^\sigma:A^\sigma --> B^\sigma of a map f:A --> B between DL's. This is where the present approach differs fundamentally from earlier work, allowing us to consider completely arbitrary maps. We want to define f^\sigma(x) and its dual f^\pi(x) for x in A^\sigma in terms of the values of f at ''nearby'' points of A. To make this idea precise, we need a topology on A^\sigma. We define \sigma, or \sigma(A^\sigma) to be the topology having as a basis the intervals [p, u] with p closed and u open. Using this topology, we define

f\sigma(x) = Ú ( Ù f(A \cap U):U in \sigma), f\pi(x) = Ù ( Ú f(A \cap U):U in \sigma).

Observe that every member a of A is an isolated point of A^\sigma because {a}= [a, a] is open. Therefore f^\sigma(a)=f(a)=f^\pi(a), i. e., f^\sigma and f^\pi are in fact extensions of f. Also, it follows from (comp) that every non-empty basic interval [p, u] contains a member of A, so A is dense in A^\sigma. Hence, f^\sigma(x) and f^\pi( x) can be thought of as the limit inferior and the limit superior of the values of f at nearby points in A.

Finally we define the canonical extension of a DLE A=(A₀, \omega^A, \omega in O) to be the DLE A^\sigma=(A₀^\sigma, (\omega^A)^\sigma, \omega in O). I. e., we set \omega^A\sigma=(\omega^A)^\sigma. The dual canonical extension A^\pi=(A₀^\delta, \omega^A\pi, \omega in O) is defined similarly. (For DL's, the notion of a canonical extension is self-dual.) There is an apparent problem with these definitions. An n-ary operation \omega^A is a map from Aⁿ into A, and the domain of (\omega^A)^\sigma is therefore A^n\sigma, not A^\sigman. However, using the obvious isomorphism between (A₀)^n\sigma and (A₀)^\sigman, we can harmlessly identify the two sets.

This completes the description of the canonical extension of a DLE. What has been accomplished is that an arbitrary DLE A has been embedded in a DLE A^\sigma of a special kind. Most notably, the DL reduct A₀^\sigma of A^\sigma has the very strong property of being doubly algebraic. Also, the way in which A sits within A^\sigma gives rise to three topologies \sigma, \sigma^\uparrow and \sigma^\downarrow on the set A^\sigma. These are defined by taking as bases intervals of the forms [p, u], [p, 1] and [0, u], respectively, with p in K(A^\sigma) and u in O(A^\sigma). Three other topologies, \iota, \iota^\uparrow and \iota^\downarrow are defined similarly, except that now p and u are required to be, respectively, compact elements and dually compact elements of the doubly algebraic lattice A₀^\sigma. The last three topologies are therefore intrinsic to the lattice A^\sigma. The topology \sigma has already been used in the definition of canonical extensions of maps and of operations. All six topologies are used in showing that many properties are preserved by canonical extensions.

This is the good news. The bad news is that some important properties are not always preserved. A spectacular example is the fact that, given a DLE homomorphism h:A --> B, the canonical extension h^\sigma:A^\sigma --> B^\sigma, is not always a homomorphism. Therefore, for the categories DLE_\muthe maps A --> A^\sigma and h --> h^\sigma for objects A and morphisms h do not constitute a functor. It is therefore important to show that for large classes of DLE's homomorphisms are preserved. Every homomorphism can of course be factored into a surjective homomorphism and an injective homomorphism. By showing that surjective homomorphisms are always preserved, the problem is reduced to the special case of injective homomorphisms, and this makes the preservation a property of the target algebra. An algebra B is said to have the property (PH) if every injective homomorphism into B is preserved, and a class of algebras is said to have (PH) if all its members have (PH). The problem then becomes: Which DLE's have (PH)? This is where Boolean products play a major role, and in the talk it will be explained how this comes about. Here we merely state one major result and two of its consequences.

Theorem. For every class K of DLE_\mu's, Var(K) has (PH) iff Pu(K) has (PH).

Corollary. Every finitely generated variety of DLE's has (PH).

Corollary. If two varieties of DLE_\mu's have (PH), then so does their lattice join.

Date received: March 20, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-87.