Derivations of MV-Algebras

In his classical paper 1 , Chang invented the notion of MV-algebra in order to provide an algebraic proof of the completeness theorem of infinite valued Lukasiewicz propositional calculus. Recently, the algebraic theory of MV-algebras is intensively studied, see 2–5 . The notion of derivation, introduced from the analytic theory, is helpful to the research of structure and property in algebraic system. Several authors 6–9 studied derivations in rings and near rings. Jun and Xin 10 applied the notion of derivation in ring and near-ring theory to BCI-algebras. In 11 , Szász introduced the concept of derivation for lattices and investigated some of its properties, for more details, the reader is referred to 9, 12–19 . In this paper, we apply the notion of derivation in ring and near-ring theory to MValgebras and investigate some of its properties. Using the notion of an isotone derivation, we characterize a derivation of MV-algebra. We introduce a new concept, called an additive derivation of MV-algebras, and then we investigate several properties. Finally, we prove that an additive derivation of a linearly ordered MV-algebra is an isotone.