Super-De Morgan functions and free De Morgan quasilattices

Abstract

A De Morgan quasilattice is an algebra satisfying hyperidentities of the variety of De Morgan algebras (lattices).In this paper we give a functional representation of the free -generated De Morgan quasilattice with two binary and one unary operations. Namely, we define the concept of super-De Morgan function and prove that the free De Morgan quasilattice with two binary and one unary operations on free generators is isomorphic to the De Morgan quasilattice of super-De Morgan functions of variables. MSC:08B05, 03C85, 06B25, 06D30, 06E30, 06B20, 08B20 Keywords Introduction and preliminariesIt is well known that the free Boolean algebra on free generators is isomorphic to the Boolean algebra of Boolean functions of variables ([7, 13, 17, 55]). The free bounded distributive lattice on free generators is isomorphic to the lattice of monotone Boolean functions of variables ([7, 13, 17, 55] [44,49,50]). In the paper [46] we have proved the completeness theorem for De Morgan functions.In this paper we give a functional representation of the free -generated De Morgan quasilattice with two binary and one unary operations. Namely, we define the concept of super-De Morgan function and prove that the free De Morgan quasilattice with two binary and one unary operations on free generators is isomorphic to the De Morgan quasilattice of super-De Morgan functions of variables. *

Hyperidentities and Related Concepts, II

“…Theorem 44 (Functional representation theorem) ( [194]) The algebra QD n is the free super-De Morgan algebra with two binary and one unary operations (with the system of free generators ∆ = {δ 1 n , . .…”

“…In particular, the investigation of hyperidentities is useful from the point of view of new technologies too, via optimization problems of block diagrams [180]. For applications of hyperidentities in discrete mathematics and topology see [61,62,74,159,190,191,192,193,194,214,307]. For characterization of Sheffer functions and primal algebras via hyperidentities see (K. Denecke, R. Pöschel [61,62]).…”