Stone duality for real-valued multisets

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In joint work with E. Dubuc and D. Mundici, the first author extended Stone duality for boolean algebras to locally finite MV-algebras. On the topological side of the duality, one has to assign to each point of a boolean space a generalized natural number by way of a multiplicity, so as to make the assignment continuous with respect to the Scott topology of the lattice of generalized natural numbers. The continuous maps between such generalized multisets are to be multiplicity-decreasing with respect to the divisibility order of generalized natural numbers. In this paper we extend these results to the class of MV-algebras that are locally weakly finite, i.e., such that all their finitely generated subalgebras split into a finite direct product of simple MV-algebras. Using the Scott topology on the lattice of subalgebras of the real unit interval (regarded with its natural MV-algebraic structure), we construct a `real-valued multiset' over the (boolean) space of maximal ideals of a locally weakly finite MV-algebra. Building on this, we obtain a duality for locally weakly finite MV-algebras that includes as a special case the above-mentioned duality for locally finite MV-algebras. We give an example that shows that the duality established in this paper via the Scott topology cannot be extended, without non-trivial modifications, to larger classes of algebras.