Abstract

Differential theory is one of the concepts has been considered by many authors since almost thirty years ago. Since that time, many articles have been appeared in this way in order to study this concept about different algebraic structures such as rings or semi-rings. In this paper, we studied the mentioned concept on algebraic structure namely semi-ring. Particularly, we introduced the notation of ( )-Derivation pair and studied some of its properties, where  is an automorphism from the semi-ring to itself. We showed that any ( )-Derivation pair is a Jordan ( )-Derivation pair but the convers is not necessary true. Furthermore, we proved that the sum of ( )-Derivation pair (resp. Jordan ( )-Derivation pair) is ( )-Derivation (resp. Jordan ( )-Derivation). Moreover, we showed that if  admits Jordan ( )-Derivation pair then its ( )-Derivation pair.