Some Examples of BL-Algebras Using Commutative Rings

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

Mathematics, 2022, vol. 10, issue 24, 1-15

Abstract: BL-algebras are algebraic structures corresponding to Hajek’s basic fuzzy logic. The aim of this paper is to analyze the structure of BL-algebras using commutative rings. Due to computational considerations, we are interested in the finite case. We present new ways to generate finite BL-algebras using commutative rings and provide summarizing statistics. Furthermore, we investigated BL-rings, i.e., commutative rings whose the lattice of ideals can be equipped with a structure of BL-algebra. A new characterization for these rings and their connections to other classes of rings is established. Furthermore, we give examples of finite BL-rings for which the lattice of ideals is not an MV-algebra and, using these rings, we construct BL-algebras with 2 r + 1 elements, r ≥ 2 , and BL-chains with k elements, k ≥ 4 . In addition, we provide an explicit construction of isomorphism classes of BL-algebras of small n size ( 2 ≤ n ≤ 5 ).

Keywords: commutative ring; BL-ring; ideal; residuated lattice; MV-algebra; BL-algebra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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