On the structure theory of Łukasiewicz near semirings. (23rd September 2017)

• Record Type: Journal Article
• Authors: Chajda, Ivan; Fazio, Davide; Ledda, Antonio
• Is Part Of: Logic journal of the IGPL. Volume 26:Number 1(2018:Feb.)
• Page Start: 14
• Page End: 28
• Publication Date: 2017-09-23
• Subjects: near semiring -- Łukasiewicz near semiring -- basic algebras -- MV-algebras -- 0-regularity -- ideals -- central elements -- decompositions -- algebraic Cantor-Bernstein theorem; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1093/jigpal/jzx044
• Languages: English
• ISSNs: 1367-0751

Abstract

Abstract: In a previous article by two of the present authors and S. Bonzio, Łukasiewicz near semirings were introduced and it was proven that basic algebras can be represented (precisely, are term equivalent to) as near semirings. In the same work it has been shown that the variety of Łukasiewicz near semirings is congruence regular. In other words, every congruence is uniquely determined by its 0-coset. Thus, it seems natural to wonder whether it could be possible to provide a set-theoretical characterization of these cosets. This article addresses this question and shows that kernels can be neatly described in terms of two simple conditions. As an application, we obtain a concise characterization of ideals in Łukasiewicz semirings. Finally, we close this article with a rather general Cantor–Bernstein type theorem for the variety of involutive idempotent integral near semirings.