State BCK-algebras and State-Morphism BCK-algebras

TL;DR

This paper introduces state and state-morphism BCK-algebras, extending BCK-algebras with a unary operator for probabilistic reasoning, and explores their structural properties and relationships.

Contribution

It defines state and state-morphism BCK-algebras, establishes their connections with measures and states, and characterizes their algebraic structure and generators.

Findings

Relation between state operators and measures on BCK-algebras

One-to-one correspondence between adjoint pairs and state-morphism operators

Generators of quasivarieties of state-morphism BCK-algebras

Abstract

In the paper, we define the notion of a state BCK-algebra and a state-morphism BCK-algebra extending the language of BCK-algebras by adding a unary operator which models probabilistic reasoning. We present a relation between state operators and state-morphism operators and measures and states on BCK-algebras, respectively. We study subdirectly irreducible state (morphism) BCK-algebras. We introduce the concept of an adjoint pair in BCK-algebras and show that there is a one-to-one correspondence between adjoint pairs and state-morphism operators. In addition, we show the generators of quasivarieties of state-morphism BCK-algebras.