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Extending Stone duality to multisets and locally finite MV-algebras. (English) Zbl 1055.06004
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 the organization of the material is exemplary. While most of the material and ideas are probably known to specialists, the reviewer considers nevertheless the paper as a valuable addition to the vast literature on Stone duality.

06D35 MV-algebras
06D50 Lattices and duality
Full Text: DOI
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