Stone-like representation theorems and three-valued filters in [formula omitted]- algebras (nilpotent minimum algebras)

Abstract

Nilpotent minimum algebras (NM-algebras) are algebraic counterpart of a formal deductive system where conjunction is modeled by the nilpotent minimum t-norm, a logic also independently introduced by Guo-Jun Wang in the mid 1990s. Such algebras are to this logic just what Boolean algebras are to the classical propositional logic. In this paper, by introducing respectively the Stone topology and a three-valued fuzzy Stone topology on the set of all maximal filters in an NM-algebra, we first establish two analogues for an NM-algebra of the well-known Stone representation theorem for a Boolean algebra, which state that the Boolean skeleton of an NM-algebra is isomorphic to the algebra of all clopen subsets of its Stone space and the three-valued skeleton is isomorphic to the algebra of all clopen fuzzy subsets of its three-valued fuzzy Stone space, respectively. Then we introduce the notions of Boolean filter and of three-valued filter in an NM-algebra, and finally we prove that three-valued filters and closed subsets of the Stone space of an NM-algebra are in one-to-one correspondence and Boolean filters uniquely correspond to closed subsets of the subspace consisting of all ultrafilters.