Topology On BCK-Modules

TL;DR

This paper introduces BCK-topological modules, explores their properties, and establishes conditions for homomorphisms to be open and continuous, advancing the understanding of topological structures in BCK-modules.

Contribution

It defines BCK-topological modules, introduces compatible and strict homomorphisms, and proves their key properties and conditions, providing a new framework in BCK-module theory.

Findings

Every decreasing sequence of submodules forms a BCK-topological module.

Strict BCK-module homomorphisms are both open and continuous.

Necessary and sufficient conditions for compatible mappings to be strict.

Abstract

In this paper, we introduce the notion of a BCK-topological module in a natural way and establish that every decreasing sequence of submodules on a BCK-module M over bounded commutative BCK-algebra X is indeed a BCK- topological module. We have defined the notion of compatible and strict BCK- module homomorphisms, and establish that a strict BCK-module homomorphism is an open as well as a continuous mapping. Also, we establish the necessary and sufficient condition for a compatible mapping to be strict.