Internal states on equality algebras

Abstract

This paper investigates properties of equality algebras introduced by Jenei as a possible algebraic semantic for fuzzy type theory. We define and study the pointed equality algebras and its subclass of compatible pointed equality algebras. We introduce and investigate the internal states and the state-morphism operators on equality algebras and on their corresponding BCK-meet-semilattices. We prove that any internal state (state-morphism) on an equality algebra is also an internal state (state-morphism) on its corresponding BCK-meet-semilattice, and we prove the converse for the case of linearly ordered equality algebras. Another main result consists of proving that any state-morphism on a linearly ordered equality algebra is an internal state on it. We show that any internal state on a linearly ordered BCK-meet-semilattice satisfying the distributivity condition is also an internal state on its corresponding equality algebra and a state-morphism on a BCK-meet-semilattice satisfying the distributivity condition is also a state-morphism on its corresponding equality algebra.