Derivations in Lukasiewicz semirings

Abstract

An axiomatization of classical propositional logic is provided by means of Boolean algebras which are term equivalent to Boolean rings. This is important because rings form a classical part of algebra whose tools can be used for the investigations. The Łukasiewicz manyvalued logic was axiomatized via so-called MV-algebras by C. C. Chang in 1950's. MV-algebras are successfully applied in the logic of quantum mechanics and hence they are considered as quantum structures. It is a natural question if also MV-algebras have their alter ego among classical structures. For this reason the so-called Łukasiewicz semirings were introduced by the first author and his collaborators in [3]-[4]. As shown, Łukasiewicz semirings are term equivalent to MV-algebras and we can use with advantage several developed tools for their study. In particular, we investigate derivations in semirings which were introduced formerly but here these semirings are enriched by an involution.