Extending Stone duality to multisets and locally finite MV-algebras.
type: Item
instance of: scholarly article
title: Extending Stone duality to multisets and locally finite MV-algebras.
zbMATH Open document ID: 1055.06004
author: Eduardo J. Dubuc
author: Daniele Mundici
author: Roberto Cignoli
published in: Journal of Pure and Applied Algebra
publication date: 2004-07-12T00:00:00Z
review text: A finite {multiset} is a pair \((X,\sigma)\) where \(X\) is finite and where \(\sigma\) assigns to every \(x\in X\) a positive natural number, the {multiplicity} of \(x\). A {morphism} between multisets \((X,\sigma)\) and \((Y,\tau)\) is a map from \(X\) to \(Y\) such that \(\tau(f(x))\) is a divisor of \(\sigma(x)\). This makes the finite multisets to a category \(\mathbf{M}\). According to the prime factorization theorem, a natural number is uniquely determined by the list of exponents of its prime divisors. A {supernatural} number is defined as a map \(\nu\) from the ordered list \(P\) of prime numbers to the set of natural numbers including \(0\) and \(\infty\). Every ordinary natural number corresponds naturally to a supernatural number \(\nu_n\). Pointwise order makes the set \({G}\) of supernatural numbers \(\nu\) to a distributive lattice, a fact often used in elementary algebra texts to prove that the natural numbers with respect to divisibility form a distributive lattice. Moreover, \(G\) is a topological space with the sets \(U_n=\{\nu\mid \nu>\nu_n\}, n=1,2,\ldots\}\) as open base. The authors define now the category \(\mathbf{C}\) where the objects are all pairs \((S,\sigma)\) where \(S\) is a Boolean space and \(\sigma\) is a continuous map from \(S\) into \(G\). The category \(\mathbf{C}\) can be construed as a category where the objects are projective limits of the objects of \(\mathbf{M}\). This goes along the lines of representing an arbitrary Boolean algebra as the direct limit of its finitely generated subalgebras and then dualizing the situation to Stone spaces. MV-algebras are a non-idempotent generalization of Boolean algebras. The formal definition of these algebras is somewhat technical but they have been around since the work of {C. C. Chang} on multivalued logic in the late fifties. The authors state as their main result that \(\mathbf{C}\) and the category of locally finite MV-algebras are co-equivalent. The paper is clearly written and t…
reviewed by: Klaus Kaiser
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 06D50
zbMATH DE Number: 2078850
zbMATH Keywords: Stone duality
zbMATH Keywords: multisets
zbMATH Keywords: MV-algebras
zbMATH Keywords: supernatural numbers
zbMATH Keywords: co-equivalence of categories
MaRDI profile type: Publication
full work available at URL: j.jpaa.2003.10.021
OpenAlex ID: W2159983957
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DOI: 10.1016/J.JPAA.2003.10.021
description: scientific article; zbMATH DE number 2078850