On MS-algebras and double MS-algebras

Abstract

In 1890 Richard Dedekind was working on a revised and enlarged edition of Dirichlet’s Vorlesungen ber Zahlentheorie (Lectures in the theory of numbers), and asked himself the following question: Given three subgroups A, B, C of an abelian group G, how many different subgroups can you get by taking intersections and sums, e.g., A+B, (A+B)∩ C, etc. In looking at this and related questions, Dedekind was led to develop the basic the- ory of lattices, which he called ”Dualgruppen”. In fact, Dedekind was ahead of his time in making the connection between modern algebra and lattice theory, and so nothing much happened in lattice theory for the next thirty years. Then, with the development of universal algebra in the thirties of the last century, Garrett Birkhoff started the general development of lattice theory. Birkhoff himself, Val`eere Glivenko, Karl Menger, John von Neumann, Oystein Ore, and others had developed enough of this new field for Birkhoff to attempt to sell it to the general mathematical community, which he did with astonishing success in the first edition of ”Lattice Theory”, [12]. Nowadays, the tentacles of lattice theory extend into algebra, analysis, topology, logic, combinatorics, linear algebra, geom- etry, category theory, probability and computer science, see [27], [31], [32], and [39].

De Morgan Stone algebras (or simply MS-algebras) have been first introduced by T.S. Blyth and J.C. Varlet as a common abstraction of de Morgan algebras and Stone alge- bras ([15], [17]) . The triple construction of MS-algebras by means of Kleene algebras and distributive lattices has been established and investigated by Blyth and Varlet in [14]. Moreover, a very interesting quadruple representation of the MS-algebras has been proved to be a very satisfactory tool in characterizing and analyzing the structure of MS- algebras, [16]. T. Katrinak added new ideas to the theory of MS-algebras via introducing