Representations of MV-algebras by sheaves

✍ Scribed by Anna R. Ferraioli; Ada Lettieri

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 198 KB
• Volume: 57
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200910116

✦ Synopsis

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.