Hyperidentities of Boolean Algebras

Abstract

In this article hyperidentities of the variety of Boolean algebras are characterized. A corollary of this characterization is the determination of a finite basis of hyperidentities for this variety. However, the class of polynomial algebras of Boolean algebras does not have a finite basis of hyperidentities. §o A hyperidentity (or V(V)-identity) is a formula in a second-order language (see [1] and [2]

Hyperidentities and Related Concepts, II

“…Theorem 34 ( [178], [174]) Any hyperidentity of the variety of all Boolean algebras is a consequence of the hyperidentities ( 39)-( 41), (17) as well as hyperidentities:…”

“…Hence, these hyperidentities are satisfied in any Boolean algebra too, by Birkhoff's subdirect representation theorem. See [174] for corresponding hyperidentities of n-ary Boolean functions. On the application of the results of [174] in modal logic see [99].…”

“…See [174] for corresponding hyperidentities of n-ary Boolean functions. On the application of the results of [174] in modal logic see [99]. 5.…”

“…On the base of these results the concept of binary G-spaces is developed in topology [74]. The hyperidentities of varieties of lattices, modular lattices, distributive lattices, Boolean algebras, De Morgan algebras and weakly idempotent lattices have been characterized in the works [174], [177], [178], [176], [195], [187], [188], [189], [202], [203], [207].…”

Hyperidentities of some generalizations of lattices

“…Therefore up to the paper of Padmanabhan and Penner [22] it was not known what are hyperidentities of (distributive) lattices. Assuming a quite different definition of a hyperidentity Movsisyan [20] has been engaged in studying the same problems as Padmanabhan and Penner [22], but also for Boolean algebras (see [19]). Padmanabhan and Penner's not easy results stimulated the author for continuation this kind of examinations in some generalizations of (distributive) lattices.…”