Representations of MV-algebras by sheaves

摘要

In this paper, inspired by methods of Bigard, Keimel, and Wolfenstein ([2]), we develop an approach to sheaf representations of MV-algebras which combines two techniques for the representation of MV-algebras devised by Filipoiu and Georgescu ([18]) and by Dubuc and Poveda ([16]). Following Davey approach ([12]), we use a subdirect representation of MV-algebras that is based on local MV-algebras. This allowed us to obtain: (a) a representation of any MV-algebras as MV-algebra of all global sections of a sheaf of local MV-algebras on the spectruum of its prime ideals; (b) a representation of MV-algebras, having the space of minimal prime ideals compact, as MV-algebra of all global sections of a Hausdorff sheaf of MV-chains on the space of minimal prime ideals, which is a Stone space; (c) an adjunction between the category of all MV-algebras and the category of MV-algebraic spaces, where an MV-algebraic space is a pair (X, F), where X is a compact topological space and F is a sheaf of MValgebras with stalks that are local.