Dualities for equational classes of Brouwerian algebras and Heyting algebras

Abstract

ABSTRACT. 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] 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].

Free nilpotent minimum algebras

“…A generalized Gödel algebra (also known in the literature as generalized linear Heyting algebra or as relative Stone algebra (see [10])) is a GMTL-algebra that satisfies the equation…”

“…Since finitely generated free generalized Gödel algebras were completely described in [10], we can have an explicit description of Free N M (X) when X is a finite set.…”

Endoprimal abelian groups

“…Now assume that A has no subgroup isomorphic to 1 2 m . Then A must have a primary component, say of exponent p k , which does not contain any subgroup isomorphic to (2 P *) 2 . In view of Corollary 2.6, in order to prove that A is not endoprimal it suffices to show that this component is not endoprimal.…”

“…Without using this name, Davey [2] proved in 1976 [2] Endoprimal abelian groups 413 that every finite chain is endoprimal as a Heyting algebra. In 1985 Davey and Werner [5] proved that the Heyting algebra 2 2 © 1 is also endoprimal, and this paper marks the appearance of the name 'endoprimal'.…”

The quest for strong dualities

“…, M)-When proving that fcj' generates M it is common to prove the following stronger statement: X\X, M') £ XQL, M) whenever X < ty [8] The quest for strong dualities 255…”