Spectral properties of cBCK-algebras

TL;DR

This paper explores the spectral properties of commutative BCK-algebras, providing new constructions, characterizations of their spectra, and functorial relationships with distributive lattices.

Contribution

It introduces a novel rooted tree construction for commutative BCK-algebras and analyzes their prime spectra, including topological and categorical properties.

Findings

Spectrum of any commutative BCK-algebra is a locally compact generalized spectral space.

Spectrum is compact iff the algebra is finitely generated as an ideal.

Involutory commutative BCK-algebras have spectra that are Priestley spaces.

Abstract

In this paper we study prime spectra of commutative BCK-algebras. We give a new construction for commutative BCK-algebras using rooted trees, and determine both the ideal lattice and prime ideal lattice of such algebras. We prove that the spectrum of any commutative BCK-algebra is a locally compact generalized spectral space which is compact if and only if the algebra is finitely generated as an ideal. Further, we show that if a commutative BCK-algebra is involutory, then its spectrum is a Priestley space. Finally, we consider the functorial properties of the spectrum and define a functor from the category of commutative BCK-algebras to the category of distributive lattices with zero. We give a partial answer to the question: what distributive lattices lie in the image of this functor?