Stone duality for real-valued multisets

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...

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Detalles Bibliográficos
Publicado:	2012
Materias:	Algebraic lattices Global sections Global sheaves Multisets MV-algebras Scott topology
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09337741_v24_n6_p1317_Cignoli http://hdl.handle.net/20.500.12110/paper_09337741_v24_n6_p1317_Cignoli
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_09337741_v24_n6_p1317_Cignoli
record_format	dspace
spelling	paper:paper_09337741_v24_n6_p1317_Cignoli2023-06-08T15:53:12Z Stone duality for real-valued multisets Algebraic lattices Global sections Global sheaves Multisets MV-algebras Scott topology 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 [0, 1] (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. © de Gruyter 2012. 2012 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09337741_v24_n6_p1317_Cignoli http://hdl.handle.net/20.500.12110/paper_09337741_v24_n6_p1317_Cignoli
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Algebraic lattices Global sections Global sheaves Multisets MV-algebras Scott topology
spellingShingle	Algebraic lattices Global sections Global sheaves Multisets MV-algebras Scott topology Stone duality for real-valued multisets
topic_facet	Algebraic lattices Global sections Global sheaves Multisets MV-algebras Scott topology
description	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 [0, 1] (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. © de Gruyter 2012.
title	Stone duality for real-valued multisets
title_short	Stone duality for real-valued multisets
title_full	Stone duality for real-valued multisets
title_fullStr	Stone duality for real-valued multisets
title_full_unstemmed	Stone duality for real-valued multisets
title_sort	stone duality for real-valued multisets
publishDate	2012
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09337741_v24_n6_p1317_Cignoli http://hdl.handle.net/20.500.12110/paper_09337741_v24_n6_p1317_Cignoli
_version_	1768546635740086272

Stone duality for real-valued multisets

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