Commutative Rings Behind Divisible Residuated Lattices

Cristina Flaut () and Dana Piciu
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Cristina Flaut: Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527 Constanţa, Romania
Dana Piciu: Faculty of Science, University of Craiova, A.I. Cuza Street, 13, 200585 Craiova, Romania

Mathematics, 2024, vol. 12, issue 23, 1-12

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 lattice of ideals can be equipped with a structure of divisible residuated lattice. We show that these rings are multiplication rings. A characterization, additional 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 ≤ n ≤ 6 ), and we give summarizing statistics.

Keywords: multiplication ring; ideal; divisible residuated lattice (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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