On monadic MV-algebras
type: Item
instance of: scholarly article
title: On monadic MV-algebras
zbMATH Open document ID: 1052.06010
author: R. Grigolia
author: Antonio Di Nola
published in: Annals of Pure and Applied Logic
publication date: 2004-08-16T00:00:00Z
review text: The variety MMV of monadic MV-algebras was introduced by Rutledge as an algebraic model for a predicate calculus of Łukasiewicz infinite-valued logic with a single variable. This variety is a generalization of Halmos's monadic Boolean algebras. For every monadic MV-algebra \(A\) it was proved by Rutledge that the set \(E\) of existential elements of \(A\) is a relatively complete subalgebra of \(A\). The authors prove that the existential part actually satisfies a stronger property, called \(m\)-relative completeness. As a consequence, they establish a one-one correspondence between monadic MV-algebras and pairs of MV-algebras \((A,E)\) where \(E\) is an \(m\)-relatively completmented subalgebra of \(A\). They study the ideal structure of MMV, and prove that MMV is congruence distributive and has the congruence extension property. Using Rutledge subdirect representation theorem, they prove that a finite monadic MV-algebra \(A\) with totally ordered existential part \(E\) is isomorphic to a product of totally ordered MV-algebras. The final section deals with a special subclass of MMV, called free cyclic. For background on MV-algebras see [\textit{R. L. O. Cignoli, I. M. L. D'Ottaviano}, and \textit{D. Mundici}, Algebraic foundations of many-valued reasoning. Kluwer Academic Publishers, Dordrecht (2000; Zbl 0937.06009)]. Rutledge's results appeared in his PhD Thesis [A preliminary investigation of the infinitely many-valued predicate calculus, Cornell University (1959)].
reviewed by: Daniele Mundici
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 08B10
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 2091342
zbMATH Keywords: MV-algebra
zbMATH Keywords: monadic MV-algebra
zbMATH Keywords: Chang algebra
zbMATH Keywords: Łukasiewicz logic
zbMATH Keywords: monadic Boolean algebra
MaRDI profile type: Publication
full work available at URL: j.apal.2003.11.031
OpenAlex ID: W1977445231
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DOI: 10.1016/J.APAL.2003.11.031
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description: scientific article; zbMATH DE number 2091342