Some remarks on infinitesimals in MV-algebras

DUBUC, EDUARDO J.; ZILBER, JORGE

Journal of Multiple-Valued Logic and Soft Computing

OLD CITY PUBLISHING INC

Año: 2016

1542-3980

Replacing f0g by the whole ideal of innitesimals yields aweaker notion of archimedean element that we call quasiarchimedean.It is known that semisimple MV-algebras with compact maximal spec-trum (in the co-Zarisky topology) are exactly the hyperarchimedeanalgebras. We characterise all the algebras with compact maximal spec-trum as being quasihyperarchimedean MV-algebras, which in a senseare non semisimple hyperarchimedean algebras. We develop some basicfacts in the theory of MV-algebras along the lines of algebraic geome-try, where innitesimals play the role of nilpotent elements, and provea MV-algebra version of Hilbert´s Nullstellensatz. Finally we considerthe relations (some inedited) between several elementary classes of MV-algebras in terms of the ideals that characterise them, and present ele-mentary (rst order with denumerable disjunctions) proofs in place ofthe set-theoretical usually found in the literature.