State pseudo equality algebras

TL;DR

This paper explores the properties of internal states and state-morphisms on pseudo equality algebras, establishing their relationships with pseudo BCK(pC)-meet-semilattices and introducing Bosbach states.

Contribution

It introduces new classes of pseudo equality algebras and proves the correspondence of states and state-morphisms between these algebras and pseudo BCK structures.

Findings

Internal states on pseudo equality algebras correspond to those on pseudo BCK(pC)-meet-semilattices.

State-morphisms on pseudo equality algebras are equivalent to those on pseudo BCK structures under certain conditions.

Bosbach states on pointed pseudo equality algebras coincide with those on their pseudo BCK counterparts.

Abstract

Pseudo equality algebras were initially introduced by Jenei and $\rm K\acute{o}r\acute{o}di$ as a possible algebraic semantic for fuzzy type theory, and they have been revised by Dvure\v censkij and Zahiri under the name of JK-algebras. The aim of this paper is to investigate the internal states and the state-morphisms on pseudo equality algebras. We define and study new classes of pseudo equality algebras, such as commutative, symmetric, pointed and compatible pseudo equality algebras. We prove that any internal state (state-morphism) on a pseudo equality algebra is also an internal state (state-morphism) on its corresponding pseudo BCK(pC)-meet-semilattice, and we prove the converse for the case of linearly ordered symmetric pseudo equality algebras. We also show that any internal state (state-morphism) on a pseudo BCK(pC)-meet-semilattice is also an internal state (state-morphism) on its corresponding pseudo equality algebra. The notion of a Bosbach state on a pointed pseudo equality algebra is introduced and proved that any Bosbach state on a pointed pseudo equality algebra is also a Bosbach state on its corresponding pointed pseudo BCK(pC)-meet-semilattice. For the case of an invariant pointed pseudo equality algebra, we show that the Bosbach states on the two structures coincide.