The word problem for Heyting algebras

Abstract

Heyting* algebras (in short HA*'s) are Heyting algebras with a unary operation * destined to capture algebraic properties of the Cantor-Bendixon derivation, and are defined by identities naturally satisfied by the open sets of every topological space, or by the ideals of every Boolean algebra. They were introduced in [6] to characterize algebraically the definable ideals in a Boolean algebra with distinguished ideals, which required some of their properties: in particular, relativization properties that generalize properties shown (for Brouwer algebras, which are dual to Heyting algebras) by McKinsey and Tarski [3]. In their article, McKinsey and Tarski established also a duality between Heyting algebras and closure algebras generated by their open elements, which can be extended to a duality between HA*'s and topological Boolean algebras generated by their open elements (see [7]). These last algebras were defined by Pierce [4], who proved many interesting properties, which thus can be translated to HA*'s. The purpose of this paper is to show that another property of Heyting algebras given by McKinsey and Tarski can be generalized to HA*'s: the finite embeddability property, which from Evans [1], implies the decidability of the word problem for HA*'s. The proof is based on a construction of subdirectly irreducible HA*~ from any HA*, which is almost as simple as the usual construction of subdirectly irreducible Heyting algebras, and which also gives all the subdirect!v irreducible HA*'s up to isomorphism. From this result and a theorem of Pierce, we finally remark that our definition of HA*'s is well suited to our first topological motivations, in the sense that each