Commutative pseudo equality algebras

TL;DR

This paper introduces and analyzes commutative pseudo equality algebras, establishing their properties, characterizations, and connections to other algebraic structures, with applications to measures, states, and valuations.

Contribution

It defines and characterizes commutative pseudo equality algebras, linking them to pseudo BCK(pC)-meet-semilattices and exploring their properties and applications.

Findings

Commutative pseudo equality algebras are distributive lattices.

Invariant pseudo equality algebras are commutative iff their pseudo BCK structures are commutative.

Finite invariant pseudo equality algebras are symmetric pseudo equality algebras.

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{c}enskij and Zahiri under the name of JK-algebras. In this paper we define and study the commutative pseudo equality algebras. We give a characterization of commutative pseudo equality algebras and we prove that an invariant pseudo equality algebra is commutative if and only if its corresponding pseudo BCK(pC)-meet-semilattice is commutative. Other results consist of proving that every commutative pseudo equality algebra is a distributive lattice and every finite invariant commutative pseudo equality algebra is a symmetric pseudo equality algebra. We also introduce and investigate the commutative deductive systems of pseudo equality algebras. As applications of these notions and results we define and study the measures and measure-morphisms on pseudo equality algebras, we prove new properties of state pseudo equality algebras, and we introduce and investigate the pseudo-valuations on pseudo equality algebras.