Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finitely Generated Varieties of Commutative BCK-algebras: Covers

Václav Cenker

TL;DR

The paper tackles the problem of describing all covers of finitely generated varieties of commutative BCK-algebras ($\mathrm{cBCK}$-algebras). It leverages the fact that finitely generated $\mathrm{cBCK}$-varieties are semisimple and that subdirectly irreducible algebras are rooted trees, reducing covers to the finite subdirectly irreducible generator $\mathbf{A}$ and its subalgebras. A key contribution is the complete description of $\mathrm{S}(\mathbf{A})$ as a union of downsets $\mathrm{S}_d(\mathbf{A})$ and a height-divisible part $\mathrm{S}_\delta(\mathrm{S}_d(\mathbf{A}))$, along with a criterion when $\mathrm{S}_\delta(\mathrm{S}_d(\mathbf{A}))$ consists of chains. The main result provides a constructive leaf-extension procedure to generate all covers: for each $\mathbf{B}\in\mathrm{S}(\mathbf{A})$ and $a\in\mathbf{B}$, extend to $\mathbf{B}_a$, identify the minimal new algebras $\mathbf{C}_a$ not in $\mathrm{S}(\mathbf{A})$, and form $\mathrm{Cov}(A)$; every cover arises in this way and covers in a finitely generated variety are themselves finitely generated. The paper also illustrates the method with explicit covers for families such as $\mathrm{V}(\mathbf{S}_n)$ and $\mathrm{V}(\mathrm{M}_P(\mathbf{S}_q))$, providing concrete recipes for constructing the lattice of finitely generated cBCK-varieties.

Abstract

The article aims at describing all covers of any finitely generated variety of cBCK-algebras. It is known that subdirectly irreducible cBCK-algebras are rooted trees (concerning their order). Also, all subdirectly irreducible members of finitely generated variety are subalgebras of subdirectly irreducible generators of that variety. The first part of the article focuses on subalgebras of finite subdirectly irreducible cBCK-algebras. In the second part of the article, a construction is presented that provides all the covers of any finitely generated variety.

Finitely Generated Varieties of Commutative BCK-algebras: Covers

TL;DR

The paper tackles the problem of describing all covers of finitely generated varieties of commutative BCK-algebras (-algebras). It leverages the fact that finitely generated -varieties are semisimple and that subdirectly irreducible algebras are rooted trees, reducing covers to the finite subdirectly irreducible generator and its subalgebras. A key contribution is the complete description of as a union of downsets and a height-divisible part , along with a criterion when consists of chains. The main result provides a constructive leaf-extension procedure to generate all covers: for each and , extend to , identify the minimal new algebras not in , and form ; every cover arises in this way and covers in a finitely generated variety are themselves finitely generated. The paper also illustrates the method with explicit covers for families such as and , providing concrete recipes for constructing the lattice of finitely generated cBCK-varieties.

Abstract

The article aims at describing all covers of any finitely generated variety of cBCK-algebras. It is known that subdirectly irreducible cBCK-algebras are rooted trees (concerning their order). Also, all subdirectly irreducible members of finitely generated variety are subalgebras of subdirectly irreducible generators of that variety. The first part of the article focuses on subalgebras of finite subdirectly irreducible cBCK-algebras. In the second part of the article, a construction is presented that provides all the covers of any finitely generated variety.
Paper Structure (4 sections, 13 theorems, 16 equations, 6 figures)

This paper contains 4 sections, 13 theorems, 16 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Subdirectly irreducible algebras and their subalgebras
  4. Covers of finitely generated varieties

Key Result

Lemma 1

Let $\mathbf{A}$ be a finite subdirectly irreducible cBCK-algebra. Then $\mathbf{A}$ is generated by $\mathrm{m}(\mathbf{A}) \cup \{a\}$, where $a$ is an atom of $\mathbf{A}$.

Figures (6)

  • Figure 1: A subdirectly irreducible cBCK-algebra and the set $A_2$ (marked with $\bullet$).
  • Figure 2: Example of cBCK-algebra satisfying $\mathrm{S(\mathbf{A})} = \mathrm{S}_d(\mathbf{A})$.
  • Figure 3: Example of particular $\mathbf{C}_a$. Dotted lines enclose $\mathbf{B}$ from which $\mathbf{B}_a$ and $\mathbf{C}_a$ arise.
  • Figure 4:
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (31)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Remark 5
  • Proposition 6
  • ...and 21 more