Every complete atomic Boolean algebra is the ideal lattice of a cBCK-algebra

TL;DR

This paper demonstrates that every complete atomic Boolean algebra can be represented as the ideal lattice of a commutative BCK-algebra, linking algebraic and topological structures through Galois connections.

Contribution

It establishes a new correspondence between complete atomic Boolean algebras and ideal lattices of cBCK-algebras, expanding the understanding of their structural relationships.

Findings

Every complete atomic Boolean algebra is the ideal lattice of a cBCK-algebra

The ideal and prime ideal structures of BCK-algebras are characterized via Galois connections

Every discrete topological space is the prime spectrum of a cBCK-algebra

Abstract

Given a complete atomic Boolean algebra, we show there is a commutative BCK-algebra whose ideal lattice is that Boolean algebra. This result is shown to exist within a larger framework involving BCK-algebras of functions, whose ideals and prime ideals are analyzed by way of a specific Galois connection. As a corollary of the main theorem, we show that every discrete topological space is the prime spectrum of a cBCK-algebra.