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Representation of topological modal algebras
by
Jacob Vosmaer
University of Amsterdam
We want to state a duality theorem for a category of topological modal algebras. So first we may ask: what is a topological algebra? Algebras (in the sense of universal algebra [BurSan2000]) are defined as structures consisting of sets and functions. One could thus topologize algebras by replacing `sets' with `topological spaces' and `functions' with `continuous functions'.
Definition Let 〈A; (fi)i ∈ I 〉 be an algebra, and let 〈A, t〉 be a topological space. We say that 〈A; (fi)i ∈ I, t〉 is a topological algebra if fi : Ani → A is continuous for the product topology on Ani. A continuous homomorphism f : 〈A; (fi)i ∈ I, tA 〉→ 〈B, (gi)i ∈ I, tB 〉 is a function that is both an algebra homomorphism and a continuous function.
In this abstract we will only consider topological algebras with compact Hausdorff topologies, i.e. compact Hausdorff algebras. To extablish a context for our main result we would like to present two questions that have been studied in the field of topological algebra.
Question 1 Given a variety V of algebras, how do the topological structure and the algebra structure interact on a (compact Hausdorff) topological algebra in V?
For instance, using compactness and continuity of ∧ one can prove that compact Hausdorff semilattices must be complete. Moreover, on compact Hausdorff lattices, the topology is uniquely determined by the lattice order, and as a corollary, a lattice homomorphism f: L→ M between compact Hausdorff lattices is continuous iff it preserves all meets and all joins [Lawson1973, CCL1980, Johnstone1982].
The second question can be motivated by the folklore fact that every Stone space is homeomorphic to a projective limit of finite sets (viewed as discrete topological spaces) [Johnstone1982, RiZa2000].
Question 2 Given a variety of algebras V, is every Stone-topological algebra A in V isomorphic to a projective limit of finite algebras in V?
We call such projective limits of finite algebras profinite algebras. This question has arisen in several different fields of algebra (see Notes of [Johnstone1982, Section VI-2]). Sufficient conditions for a positive answer to Question 2 are for V to have equationally definable principal congruences, or for V to be finitely generated [CDFJ2004].
The answers to these two questions become entwined if one considers Boolean algebras. The reason for that lies in the fact that it follows from a central result in Pontryagin duality that every compact Hausdorff Boolean algebra is Stone (i.e. is zero-dimensional) [Strauss1968]. As regards the second question, it is well-established that every Stone Boolean algebra is profinite [Numakura1957]. Moreover, these categories of Boolean algebras can also be characterized lattice-theoretically as complete atomic Boolean algebras. We summarize this below.
Fact([Lawson1973, Johnstone1982]) Pro-BAf ≅ StoneBA = KHausBA ≅ CABA.
Below we will see that if we consider modal algebras [Venema2007] instead of Boolean algebras then the above equivalences no longer hold. By KFr we denote the category of Kripke frames and p-morphisms. Additionally, by ImFKFr we denote the full subcategory of image-finite Kripke frames, i.e. those frames in which each state has only finitely many successors. Our main result is the following duality:
Theorem KHausMA ≅ ImFKFrop.
Recall that KFr is dually equivalent to CAMA, the category of complete atomic modal algebras with completely additive modal operators and complete homomorphisms [Thomason1975]. Combined with the duality for profinite modal algebras [Vosmaer2006], this leads to the following result.
Corollary Pro-MAf ⊆ StoneMA = KHausMA ⊆ CAMA, where both inclusions are full and strict.
Date received: June 9, 2008
Copyright © 2008 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 # caxi-34.