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

C. Matthew Evans

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.

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.

Paper Structure

This paper contains 4 sections, 13 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Main Theorem
  3. BCK-algebras of functions
  4. A small application

Key Result

Proposition 1.3

A BCK-algebra $\mathbf{A}$ is simple if and only if, for any $x,y\in \mathbf{A}$ with $y\neq 0$, there is a natural number $n$ such that $x\boldsymbol{\cdot} y^n=0$.

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 23 more