Locally finite MV-algebras, multisets and a conjecture of António Aniceto Monteiro

CIGNOLI, ROBERTO LEONARDO OSCAR

Lisboa

Congreso; Colóquio António Aniceto Monteiro; 2007

Sociedade Matemática Portuguesa

:  In [1], Stone duality between Boolean algebras and inverse limits of finite sets is extended to a duality between locally finite MV-algebras and a category of multisets naturally arising as inverse limits of finite multisets. I shall show how this duality specializes to a duality between MVn, the subvariety of MV-algebras generated by the n-element Łukasiewicz chain, and certain Boolean spaces with a distinguished family of closed subsets, for each 2 ≤ n Îw [2, 5].  I will comment the role played by this duality in [4], where a conjecture of A. A. Monteiro on the structure of maximal subalgebras of algebras in MV3 is proved, and in [1], where a description of free algebras in MVn over sets of free generators of arbitrary cardinal is given, generalizing a result of A. A. Monteiro for finite sets of free generators. References: [1] M. Busniche, R. Cignoli, Free MVn-algebras, submitted. [2] R. Cignoli, Natural dualities for the algebras of Łukasiewicz finitely-valued logics (Abstract), Bulletin of Symbolic Logic, 2 (1996), p. 218. [3] R. Cignoli, E. J. Dubuc, D. Mundici, Extending Stone duality to multisets and locally finite MV-algebras, Journal of Pure and Applied Algebra, 189 (2004) 37 – 59. [4] R. Cignoli, L. Monteiro, Maximal Subalgebras of MVn-algebras. A Proof of a Conjecture of A. Monteiro, Studia Logica, 84 (2006), 393--495. [5] P. Niederkorn, Natural dualities for varieties of MV-algebras, Journal of Mathematical Analysis and Applications, 225 (2001), 58--73.