States on semi-divisible generalized residuated lattices reduce to states on MV-algebras
type: Item
instance of: scholarly article
title: States on semi-divisible generalized residuated lattices reduce to states on MV-algebras
zbMATH Open document ID: 1188.06009
author: Janne Mertanen
author: Esko Turunen
published in: Fuzzy Sets and Systems
publication date: 2009-08-28T00:00:00Z
review text: The authors continue their study of the set MV\((L)\) of complemented elements of a residuated lattice \(L\) [Soft Comput. 12, No.~4, 353--357 (2008; Zbl 1138.06006)]. Such structures are related to mathematical fuzzy logic as well as to extended probability theory. In the present paper they deal with good generalized (or pseudo) residuated lattices (the underlying monoidal operation need not be commutative, \(x^{-\sim} = x^{\sim -}\), and then MV\((L)\) = \(\{x\in L: x^{-\sim} = x^{\sim -} = x\}\) = \(\{x^-: x\in L\}\) = \(\{x^\sim: x\in L\}\)). Theorem 12 states that those good residuated lattices whose complemented elements induce a pseudo-MV-algebra are exactly semi-divisible (a condition weaker that divisibility) generalized residual lattices. Further, they study maximal filters on a semi-divisible generalized residuated lattice \(L\) and their relationships to maximal filters on MV\((L)\). Finally, they show (as stated in the title) that Riečan and Bosbach states reduce to states on MV-algebras.
reviewed by: Roman Frič
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 81P10
zbMATH DE Number: 5599504
zbMATH Keywords: good generalized residuated lattice
zbMATH Keywords: complemented element
zbMATH Keywords: semi-divisibility
zbMATH Keywords: pseudo-MV-algebra
zbMATH Keywords: maximal filter
zbMATH Keywords: Riečan state
MaRDI profile type: Publication
full work available at URL: j.fss.2008.01.036
OpenAlex ID: W2052933125
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DOI: 10.1016/J.FSS.2008.01.036
description: scientific article; zbMATH DE number 5599504