Multipotent GBL-algebras
type: Item
instance of: scholarly article
title: Multipotent GBL-algebras
zbMATH Open document ID: 1206.06009
author: Anatolij Dvurečenskij
author: Tomasz Kowalski
published in: Algebra Universalis
publication date: 2011-01-07T00:00:00Z
review text: GBL-algebras are residuated lattices satisfying a divisibility law. An element of a GBL-algebra \(\mathbf{A}\) is \(n\)-potent if \(a^{n+1}=a^{n}\) holds. Thus, \(\mathbf{A}\) is \(n\)-potent if there is an \(n\in \omega\) such that every \(a\in A\) is \(n\)-potent. An algebra is called multipotent if every element is \(n\)-potent for some \(n\), but the algebra itself is not \(n\)-potent for any \(n\). Every finite GBL-algebra is \(n\)-potent for some \(n\), and every \(n\)-potent algebra is multipotent, but neither of the converses holds. A GBL-algebra is normal if every filter of \(\mathbf{A}\) is normal. The aim of this paper is to present some properties of the multipotent GBL-algebras. First, we have that every multipotent GBL-algebra \(\mathbf{A}\) is normal, and if \(\mathbf{A}\) is subdirectly irreducible then it contains a greatest idempotent element \(a<1\). Next, it is shown that subdirectly irreducible multipotent GBL-algebras with a unique idempotent element \(a<1\) are pseudo-MV-algebras. More precisely, they become pseudo-MV-algebras when expanded with a constant interpreted as the bottom element. If \(\mathbf{A}\) is a subdirectly irreducible multipotent GBL-algebras, then \(\mathbf{A}\) is decomposable as ordinal sum. Also, the authors slightly extend the results of \textit{P. Jipsen} and \textit{F. Montagna} [ibid. 55, No.~2--3, 227--238 (2006; Zbl 1109.06011); ibid. 60, No.~4, 381--404 (2009; Zbl 1192.06011)] on conditions implying commutativity (these results explicitly state that any \(n\)-potent GBL-algebra is commutative). In this sense we have that the variety of GBL-algebras generated by its multipotent members is commutative. This implies that every multipotent GBL-algebra satisfying preliniarity is commutative. But if the variety of integral GBL-algebras is noncommutative then it is not generated by its multipotent members. Finally, the authors explain what happens with the results presented here in other (sub)varieties: FL, pseudo BL-alg…
reviewed by: Florentina Chirteş
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 06F15
zbMATH DE Number: 5833771
zbMATH Keywords: GBL-algebra
zbMATH Keywords: Archimedean \(l\)-group
zbMATH Keywords: \(n\)-potency
zbMATH Keywords: commutativity
describes a project that uses: Pseudo Hoops
MaRDI profile type: Publication
full work available at URL: s00012-010-0085-0
OpenAlex ID: W1998275836
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DOI: 10.1007/S00012-010-0085-0
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description: scientific article; zbMATH DE number 5833771