Semisimplicity and the discriminator in bounded BCK-algebras
type: Item
instance of: scholarly article
title: Semisimplicity and the discriminator in bounded BCK-algebras
zbMATH Open document ID: 1201.06015
published in: Algebra Universalis
publication date: 2010-09-08T00:00:00Z
review text: The purpose of this paper is to characterize certain types of subclasses of the class bBCK of bounded BCK-algebras. The first main theorem of the article states that a relative subvariety \(V\) of bBCK is semisimple iff it satisfies a certain sequence of arithmetic properties, and, moreover, in this case, \(V\) is a variety. The second main theorem of the article states that, given a relative subvariety \(V\) of bBCK, the following equivalences hold: \(V\) is a discriminator variety iff \(V\) is semisimple and congruence-permutable iff \(V\) is a semisimple relative subvariety of involutive BCK-algebras. The author also extends these two main theorems to strong expansions of bBCK and concludes by an example of an involutive BCK-algebra that is not congruence-permutable.
reviewed by: Claudia Mureşan
Mathematics Subject Classification ID: 06F35
Mathematics Subject Classification ID: 08B99
zbMATH DE Number: 5781183
zbMATH Keywords: bounded BCK-algebras
zbMATH Keywords: semisimple relative subvariety
zbMATH Keywords: discriminator variety
zbMATH Keywords: strong expansion
author: Antoni Torrens
MaRDI profile type: Publication
full work available at URL: s00012-010-0061-8
OpenAlex ID: W2006766521
cites work: Algebraizable logics
cites work: On the quasivariety of BCK-algebras and its subvarieties
cites work: Varieties of commutative residuated integral pomonoids and their residuation subreducts
cites work: ASSERTIONALLY EQUIVALENT QUASIVARIETIES
cites work: Q3934450
cites work: Glivenko like theorems in natural expansions of BCK-logic
cites work: Protoalgebraic logics
cites work: Residuated lattices. An algebraic glimpse at substructural logics
cites work: Bounded BCK-algebras and their generated variety
cites work: Boolean representation of bounded BCK-algebras
cites work: PREDICATE AND SET-THEORETIC CALCULI BASED ON LOGIC WITHOUT CONTRACTIONS
cites work: Impossibility of defining the class of \(L_ 0\)-algebras by means of identities
cites work: Q3218120
cites work: Q3218122
cites work: Q4115148
cites work: Semisimplicity, EDPC and discriminator varieties of residuated lattices
cites work: Q3932964
cites work: Finite Basis Theorems for Relatively Congruence-Distributive Quasivarieties
cites work: On The Role of The Polynomial (X → Y) →Y in Some Implicative Algebras
cites work: An approach to Glivenko's theorem in algebraizable logics
cites work: On varieties of biresiduation algebras
cites work: Q3666901
cites work: Q3666903
DOI: 10.1007/S00012-010-0061-8
description: scientific article; zbMATH DE number 5781183