Dualities for Equational Classes of Brouwerian Algebras and Heyting Algebras

This paper focuses on the equational class S„ of Brouwerian algebras and the equational class L„ of Heyting algebras generated by an »-element chain. Firstly, duality theories are developed for these classes. Next, the projectives in the dual categories are determined, and then, by applying the dualities, the injectives and absolute subretracts in Sn and L„ are characterized. Finally, free products and the finitely generated free algebras in S„ and L„ are described. Recently there has been considerable interest in distributive pseudocomplemented lattices, Brouwerian algebras and Heyting algebras. In particular, activity has centered around the equational subclasses ([8], [11], [24], [35], [36]), and steps have been made towards the determination of the injectives, absolute subretracts, free products and free algebras in these classes ([1], [2], [3], [12], [19], [20], [21], [27], [31], [32], [33], [34], [46], [47] ). In this work attention is focused upon the equational class S„ of Brouwerian algebras and the equational class Ln of Heyting algebras generated by an n-element chain. Firstly, a duality theory is developed for each of these classes, the dual of an algebra being a Boolean space endowed with a continuous action of the endomorphism monoid of the n-element chain. Next, the projectives in the dual categories are determined, and then, by applying the dualities, the injectives and absolute subretracts in S„ and L„ are characterized. Finally, free products and the finitely generated free algebras in S„ and Ly, are described. 1. The categories. Our standard references on category theory, universal algebra, and lattice theory are S. Mac Lane [37], G. Grätzer [17], and G. Gra'tzer [18] respectively; for our general topological requirements we refer to J. Dugundji [13] and for a discussion of Boolean a spaces we call on P. R. Halmos [23]. Received by the editors August 13, 1974. AMS (MOS) subject classifications (1970). Primary 06A35, 18A40, 54H10; Secondary 02C0S, 08A10, 08A25, 18C0S, 54F05, 54G05.