Some examples of BL-algebras using commutative rings

TL;DR

This paper explores the structure of BL-algebras through commutative rings, focusing on finite cases, new generation methods, and the characterization of BL-rings, including their lattice of ideals and construction of specific finite BL-algebras.

Contribution

It introduces new methods to generate finite BL-algebras from commutative rings and characterizes BL-rings, expanding understanding of their lattice structures and examples.

Findings

Presented new generation techniques for finite BL-algebras.

Characterized BL-rings and their ideal lattices.

Constructed specific finite BL-algebras and BL-chains.

Abstract

The aim of this paper is to analize the structure of BL-algebras using commutative rings. From computational considerations, we are very interested in the finite case. We present new ways to generate finite BL-algebras using commutative rings and we give summarizing statistics. Furthermore, we investigated BL-rings, i.e., commutative rings whose the lattice of ideals can be equiped with a structure of BL-algebra. A new characterization for these rings and their connections to other classes of rings are established. Also, we give examples of finite BL-rings for which their lattice of ideals is not an MV-algebra and using these rings we construct BL-algebras with $2^{r}+1$ elements, $r\geq 2,$ and \ all BL-chains with $k$ elements, $k\geq 4.$