The Derivative Theory On BL - Algebras And MV - Algebras

The notion of derivations is derived from the analytic theory, which is helpful for studying of algebraic systems. In this paper, we will investigate the theory of derivation in BL-algebras、MV-algebras and hyper MV-algebras. This paper is organized as follows:Firstly, we introduce the notions of ☉-derivations and strong ☉-derivations in BL-algebras and discuss some properties. We prove that the fixed point set Fd(A)={x ∈ A:dx = x} of d is down-closed set in a BL-algebra A for the strong ☉-derivation d. Moreover, we introduce the principal ☉-derivations da:da(x) = a ☉ x and prove that Fda(A) is the lattice ideal of A. Finally, we characterize a Godel algebra and a linear Godel algebra by the fixed point set of ☉-derivation, respectively.Secondly, using the endomorphism of MV-algebra, we introduce the notions of f derivations and g derivations of MV-algebras, respectively. And some related properties are investigated. Moreover, the set Fd(M)f of all fixed points for an isotone f derivation d is proved to be an ideal of M and the set Fd(M)g of all fixed points for a strong g derivation d is proved to be an ideal of M. Furthermore, characterizations of Boolean algebras and linear Boolean algebras are derived by properties of g derivations, respectively. Finally, the relationships between f derivations and g derivations are discussed.At last, according to the characterizations of hyperstructure, we define derivations and strong derivations in hyper MV-algebras and discussed some basic properties of them. Using the notion of the strong derivation, we give some characterizations of a derivation of an hyper MV-algebra. Moreover, hyper MV-ideals are derived by properties of derivations.