On the structure of generalized BL-algebras
type: Item
instance of: scholarly article
title: On the structure of generalized BL-algebras
zbMATH Open document ID: 1109.06011
author: Franco Montagna
author: Peter Jipsen
published in: Algebra Universalis
publication date: 2007-01-29T00:00:00Z
review text: A residuated lattice is an algebra \(L=(L,\wedge ,\vee ,\cdot ,e,\backslash ,/)\) such that \((L,\wedge ,\vee )\) is a lattice, \((L,\cdot ,1)\) is a monoid and multiplication is both left and right residuated, with \(\backslash \) and \(/\) as residuals, i.e. \(a\cdot b\leq c\Leftrightarrow a\leq c/b\Leftrightarrow b\leq a\backslash c\) for all \(a,b,c\in L.\) A generalized BL-algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities \(x\wedge y=((x\wedge y)/y)y=y(y\backslash (x\wedge y)).\) In the first part of this paper the authors shown that all finite GBL-algebras are commutative, hence they can be constructed by iterating ordinal sums and direct products of Wajsberg hoops. By observing that the idempotents in GBL-algebra form a subalgebra of elements that commute with all other elements, subsequently they construct subdirectly irreducible noncommutative integral GBL-algebras that are not ordinal sums of generalized MV-algebras. Also, they give equational bases for the varieties generated by such algebras; this construction provides a new way of order-embedding the lattice of varieties of \(l\)-groups into the lattice of varieties of integral GBL-algebras. We remark that the results of this paper also apply to pseudo BL-algebras.
reviewed by: Dana Piciu
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 5119634
zbMATH Keywords: generalized BL-algebra
zbMATH Keywords: basic logic
zbMATH Keywords: residuated lattice
MaRDI profile type: Publication
full work available at URL: s00012-006-1960-6
OpenAlex ID: W2027528905
DOI: 10.1007/S00012-006-1960-6
description: scientific article; zbMATH DE number 5119634