Commutative rings behind divisible residuated lattices

TL;DR

This paper explores the relationship between divisible residuated lattices and commutative rings, showing that such rings are multiplication rings and analyzing their structure through finite examples and explicit constructions.

Contribution

It characterizes rings whose ideal lattices form divisible residuated lattices, establishing their connection to multiplication rings and providing explicit classifications for small cases.

Findings

Rings with ideal lattices as divisible residuated lattices are multiplication rings.

Explicit classifications of small divisible residuated lattices are provided.

Connections between these lattices and other ring classes are established.

Abstract

Divisible residuated lattices are algebraic structures corresponding to a more comprehensive logic than Hajek's basic logic with an important significance in the study of fuzzy logic. The purpose of this paper is to investigate commutative rings whose the lattice of ideals can be equipped with a structure of divisible residuated lattice. We show that these rings are multiplication rings. A characterization, more examples and their connections to other classes of rings are established. Furthermore, we analyze the structure of divisible residuated lattices using finite commutative rings. From computational considerations, we present an explicit construction of isomorphism classes of divisible residuated lattices (that are not BL-algebras) of small size n $(2 \le n \le 6)$ and we give summarizing statistics.