EQ-algebras with internal states

Abstract

The main goal of this paper is to investigate EQ-algebras with internal states and state morphism good EQ-algebras. To begin with, we introduce the notion of EQ-algebras with internal states (simplify, SEQ-algebras) and discuss the relation between SEQ-algebras and state EQ-algebras. In the following, we study state filters (simplify, S-filters) and state prefilters (simplify, S-prefilters) of SEQ-algebras and discuss subdirectly irreducible SEQ-algebras. We focus on algebraic structures of the set SPF $$(E,\sigma )$$ of all S-prefilters on a SEQ-algebra and obtain that SPF $$(E,\sigma )$$ forms a complete Brouwerian lattice, when E is an $$\ell $$ EQ-algebra or good. Moreover, for $$\ell $$ EQ-algebras, SPF $$(E,\sigma )$$ forms a Heyting algebra if $$\sigma $$ is faithful and preserves $$\rightarrow $$ . Then, we introduce the $$\sigma $$ -co-annihilator of a non-empty set A on a SEQ-algebra. As applications, we give a characterization for minimal prime S-prefilters of state morphism good EQ-algebras and characterize the representable state morphism good EQ-algebras by minimal prime S-prefilters.