Monadic MV-algebras are equivalent to monadic \(\ell \)-groups with strong unit
type: Item
instance of: scholarly article
title: Monadic MV-algebras are equivalent to monadic \(\ell \)-groups with strong unit
zbMATH Open document ID: 1247.06008
published in: Studia Logica
publication date: 2012-03-09T00:00:00Z
review text: Monadic MV-algebras were introduced by J. D. Rutledge as an algebraic model for the (monadic) predicate calculus of Łukasiewicz infinite-valued logic with a single individual variable. These algebras have been studied by many authors, such as Di Nola, Grigolia, Belluce, Lettieri. It is well known, by Mundici's famous theorem, that the MV-algebras are categorically equivalent to the \(\ell\)-groups with strong unit. More precisely, if \(G\) is an abelian \(\ell\)-group and \(u\) is a strong unit of \(G\), then the interval \([0, u]\) can be endowed with the structure of an MV-algebra, denoted \((\Gamma, u)\). The functor \(\Gamma\), defined in this way, establishes the above categorical equivalence. The aim of the present paper is to extend Mundici's functor \(\Gamma\) to the category of monadic MV-algebras. In order to do that, the authors define quantifiers on the lattice monoid of good sequences \({M_A}\) of an MV-algebra \(A\), introduce the notion of monadic \(\ell\)-group, and construct the monadic Chang \(\ell\)-group. Namely, if \({M_{\exists A}}\) denotes the lattice monoid of good sequences of elements of \(\exists A\), then define: \(\exists_M, \forall_M:M_A \longrightarrow M_A\) by \(\exists_M(M_A)=M_{\exists A}\) and \(\forall_M a=\neg \exists_M \neg a\) for all \(a\in M_A\). Quantifiers on Chang's \(\ell\)-group \(G_A\) of a monadic MV-algebra \(A\) are defined as follows: \(\exists_G, \forall_G:{G_A} \rightarrow {G_A}\) by \(\exists_G([\bar a, \bar b])=[\exists_M((\bar a\vee \bar b)-\bar b), \forall_M((\bar a \vee \bar b)-\bar a)]\) and \(\forall_G([\bar a, \bar b])=\neg \exists_G \neg [\bar a, \bar b]\). If \((G; +, -, 0, \leq)\) is an \(\ell\)-group and \(u>0\) is a fixed element of \(G\), then the authors define the notion of monadic \(\ell\)-group \({G}=(G; +, -, 0, \leq, u, \exists)\). For a monadic MV-algebra \({A}\), the structure \((G_A; [(1), (0)], \exists_G)\) is a monadic \(\ell\)-group, which is called the monadic Chang \(\ell\)-group. The …
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 06F15
zbMATH DE Number: 6013466
zbMATH Keywords: monadic MV-algebra
zbMATH Keywords: monadic \(\ell\)-group
zbMATH Keywords: monadic Chang \(\ell\)-group
zbMATH Keywords: natural equivalence
author: Cecilia Rossana Cimadamore
author: Patricio Díaz Varela
reviewed by: Lavinia Ciungu
MaRDI profile type: Publication
full work available at URL: s11225-011-9332-4
OpenAlex ID: W2033476414
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DOI: 10.1007/S11225-011-9332-4
description: scientific article; zbMATH DE number 6013466