On a definition of a variety of monadic \(\ell\)-groups.
type: Item
instance of: scholarly article
title: On a definition of a variety of monadic \(\ell\)-groups.
zbMATH Open document ID: 1337.06011
author: J. L. Castiglioni
author: R. A. Lewin
author: Marta S. Sagastume
published in: Studia Logica
publication date: 2014-06-19T00:00:00Z
review text: In the paper by the first author, \textit{M. Menni}, and \textit{M. Sagastume} [Stud. Log. 90, No. 1, 93-124 (2008; Zbl 1165.06002)], it was proved that the category \(\mathrm{IRL}_0\) of integral residuated lattices with bottom is equivalent to a category whose objects are \(c\)-differential residuated lattices satisfying a certain technical condition. The corresponding adjunction extends the one between Heyting algebras and Nelson algebras obtained in the paper by \textit{R. Cignoli} [Algebra Univers. 23, 262-292 (1986; Zbl 0621.06009)]. In his paper [J. Funct. Anal. 65, 15-63 (1986; Zbl 0597.46059)], the present reviewer established a categorical equivalence \(\Gamma\) between MV-algebras and lattice ordered abelian groups with a distinguished (strong, order) unit, for short, unital \(l\)-groups. MV-algebras are the algebras of Łukasiewicz infinite-valued logic. Unital \(l\)-groups provide a modern reformulation of Euclidean magnitudes equipped with a distinguished Archimedean unit of measurement. In the last sections of the paper under review, the authors specialize their results to the case of MV-algebras and the corresponding category of monadic MV-algebras induced by ``Kalman's functor'' \(K^*\). Moreover, they extend the construction to \(l\)-groups introducing the new category of monadic \(l\)-groups together with a functor paralleling the \(\Gamma\) functor.
reviewed by: Daniele Mundici
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 03G25
zbMATH DE Number: 6305275
zbMATH Keywords: MV-algebras
zbMATH Keywords: monadic \(\ell\)-groups
zbMATH Keywords: residuated lattices
MaRDI profile type: Publication
full work available at URL: s11225-012-9464-1
OpenAlex ID: W2089472746
cites work: Q4083730
cites work: On some categories of involutive centered residuated lattices
cites work: The class of Kleene algebras satisfying an interpolation property and Nelson algebras
cites work: Quantifiers on distributive lattices
cites work: On monadic MV-algebras
cites work: Q3282928
cites work: The algebra of topology
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DOI: 10.1007/S11225-012-9464-1
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description: scientific article; zbMATH DE number 6305275