New View of Homomorphic Properties of BCK-Algebra in Terms of Some Notions of Discrete Dynamical System

Abstract

In the present manuscript, we introduce the concept of a discrete dynamical system (Ⱬ, 𝛹) in BCK-algebra where Ⱬ is a BCK-algebra and 𝛹 is a homomorphism from Ⱬ to Ⱬ and establish some of their related properties. We prove that the set of all fixed points and the set of all periodic points in BCK-algebra Ⱬ are the BCK-subalgebras. We show that when a subset of BCK-algebra Ⱬ is invariant concerning 𝛹. We prove that the set of all fixed points and the set of all periodic points in commutative BCK-algebra Ⱬ with relative cancellation property are the ideals of Ⱬ. We also prove that the set of all fixed points in Ⱬ is an S-invariant subset of a BCK-algebra Ⱬ.