Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On pseudo-irreducibility and Boolean lifting property of filters in residuated lattices

Esmaeil Rostami

TL;DR

The paper investigates when filters in a residuated lattice admit Boolean lifting from quotients, introducing pseudo-irreducible filters and relating them to the Boolean lifting property (BLP). It develops several equivalent characterizations of BLP, analyzes the residuated lattice of fractions to transfer BLP between $L$ and $L[S]$, and studies the radical of filters via Rad(F) and the transitional property of radicals decomposition (TPRD). It shows that weak MTL-algebras and their generalizations satisfy TPRD, ensuring Rad(L) has BLP, and provides a topological criterion for Rad(L) to have BLP in terms of clopen subsets of the Stone space of maximal filters. Overall, the work bridges algebraic and topological viewpoints to deepen understanding of the Boolean lifting property in residuated lattices and addresses open questions about radicals.

Abstract

In this paper, we introduce the notion of a pseudo-irreducible filter in a residuated lattice and compare this concept with related notions such as prime and maximal filters. Then, we recall the Boolean lifting property for filters and present useful characterizations for this property using pseudo-irreducible filters and the residuated lattice of fractions. Next, we study the Boolean lifting property of the radical of a filter. Furthermore, we introduce weak MTL-algebras and residuated lattices that have the transitional property of radicals decomposition (TPRD) as generalizations of several algebraic structures, including Boolean algebra, MV-algebra, BL-algebra, MTL-algebra, and Stonean residuated lattice. Moreover, by comparing weak MTL-algebras with other classes of residuated lattices, we address an open question concerning the Boolean lifting property of the radical of a residuated lattice. Finally, we give a topological answer to an open question about the Boolean lifting property of the radical of a residuated lattice. Several additional results are also obtained, further enriching the understanding of the Boolean lifting property in residuated lattices.

On pseudo-irreducibility and Boolean lifting property of filters in residuated lattices

TL;DR

The paper investigates when filters in a residuated lattice admit Boolean lifting from quotients, introducing pseudo-irreducible filters and relating them to the Boolean lifting property (BLP). It develops several equivalent characterizations of BLP, analyzes the residuated lattice of fractions to transfer BLP between and , and studies the radical of filters via Rad(F) and the transitional property of radicals decomposition (TPRD). It shows that weak MTL-algebras and their generalizations satisfy TPRD, ensuring Rad(L) has BLP, and provides a topological criterion for Rad(L) to have BLP in terms of clopen subsets of the Stone space of maximal filters. Overall, the work bridges algebraic and topological viewpoints to deepen understanding of the Boolean lifting property in residuated lattices and addresses open questions about radicals.

Abstract

In this paper, we introduce the notion of a pseudo-irreducible filter in a residuated lattice and compare this concept with related notions such as prime and maximal filters. Then, we recall the Boolean lifting property for filters and present useful characterizations for this property using pseudo-irreducible filters and the residuated lattice of fractions. Next, we study the Boolean lifting property of the radical of a filter. Furthermore, we introduce weak MTL-algebras and residuated lattices that have the transitional property of radicals decomposition (TPRD) as generalizations of several algebraic structures, including Boolean algebra, MV-algebra, BL-algebra, MTL-algebra, and Stonean residuated lattice. Moreover, by comparing weak MTL-algebras with other classes of residuated lattices, we address an open question concerning the Boolean lifting property of the radical of a residuated lattice. Finally, we give a topological answer to an open question about the Boolean lifting property of the radical of a residuated lattice. Several additional results are also obtained, further enriching the understanding of the Boolean lifting property in residuated lattices.
Paper Structure (5 sections, 40 theorems, 19 equations)

This paper contains 5 sections, 40 theorems, 19 equations.

Table of Contents

  1. Preliminaries
  2. Pseudo-irreducible filters in residuated lattices
  3. Boolean Lifting Property and pseudo-irreducibility
  4. Boolean Lifting Property and the residuated lattice of fractions
  5. Boolean Lifting Property and the radical of filters

Key Result

Proposition 1.2

[7, 11, 21] Let $L$ be a residuated lattice, $x, y, z\in L$ and $e, f\in B(L)$. Then we have the following statements:

Theorems & Definitions (57)

  • Definition 1.1
  • Proposition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Definition 2.1
  • Example 2.2
  • ...and 47 more