Every complete atomic Boolean algebra is the ideal lattice 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.

Published in the European Journal of Mathematics (volume 10, June 2024)