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Commutative rings behind divisible residuated lattices

Cristina Flaut, Dana Piciu

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.

Commutative rings behind divisible residuated lattices

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 and we give summarizing statistics.

Paper Structure

This paper contains 5 sections, 13 theorems, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Divisible residuated lattices and Multiplication rings
  4. Examples of divisible residuated lattices using commutative rings
  5. Conclusions

Key Result

Proposition 4

([I; 09]) Let $(L,\vee ,\wedge ,\odot ,\rightarrow ,0,1)$ be a residuated lattice. Then we have the equivalences: (i) $L\ $satisfies $(\func{div})$ condition; (ii) $\ \ $For all $x,y\in L$ if $y\leq x$ then there exists $z\in L$ such that $y=z\odot x.$

Figures (2)

  • Figure 1: Latices with five elements.
  • Figure 2: Latices with five elements.

Theorems & Definitions (33)

  • Definition 1
  • Example 2
  • Definition 3
  • Proposition 4
  • Example 5
  • Lemma 6
  • Definition 7
  • Theorem 8
  • Proposition 9
  • Proposition 10
  • ...and 23 more