Commutative pseudo-equality algebras

Pseudo-equality algebras were initially introduced by Jenei and Kóródi as a possible algebraic semantic for fuzzy-type theory, and they have been revised by Dvurečenskij 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 introduce the notion of a commutative deductive system of a pseudo-equality algebra, and we give equivalent conditions for this notion. 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. We prove that any measure morphism on a pseudo-equality algebra is a measure on it, and the kernel of a measure is a commutative deductive system. We show that the quotient pseudo-equality algebra over the kernel of a measure is a commutative pseudo-equality algebra. It is also proved that a pseudo-equality algebra possessing an order-determining system is commutative. We prove that the two types of internal states on a pseudo-equality algebra coincide if and only if it is a commutative pseudo-equality algebra. Given a pseudo-equality algebra A, it is proved that the kernel of a commutative pseudo-valuation on A is a commutative deductive system of A. If, moreover, A is commutative, then we prove that any pseudo-valuation on A is commutative.