Some Remarks on Infinitesimals in MV-algebras
Eduardo J. Dubuc and Jorge C. Zilber
Replacing {0} by the whole ideal of infinitesimals yields a weaker notion of archimedean element that we call quasiarchimedean. It is known that semisimple MV-algebras with compact maximal spectrum (in the co-Zarisky topology) are exactly the hyperarchimedean algebras. We characterise all the algebras with compact maximal spectrum as being quasihyperarchimedean MV-algebras, which in a sense are non semisimple hyperarchimedean algebras. We develop some basic facts in the theory of MV-algebras along the lines of algebraic geometry, where infinitesimals play the role of nilpotent elements, and prove a MV-algebra version of Hilbert’s Nullstellensatz. Finally we consider the relations (some inedited) between several elementary classes of MV-algebras in terms of the ideals that characterise them, and present elementary (first order with denumerable disjunctions) proofs in place of the set-theoretical usually found in the literature.
Keywords: MV-algebras; infinitesimals; nullstellensatz; hyperarchimedian