Indian Journal of Science and Technology

Abstract

Objectives: Semirings is an important algebraic structure with applications in theory of automata, formal languages and theoretical computer science. The mappings which enforces commutativity in semirings remains attractive for researchers, since commutativity would be helpful in calculations and bring it's applications to ease. Our aim is to enforce commutativity in semirings by generalizing the classical theorem of Martindale [14, Theorem 3] with generalized m-derivation. Further, we discuss that composition of two generalized m-derivations ensure that one of their associated derivation must be trivial. Method: We use generalized m-derivations which is associated to multiplicative derivations in certain semirings. Findings: We find the conditions of commutativity in semirings through these particular generalized m-derivations. Moreover, we discuss the characteristics of these mappings in weakly cancellative semirings. Novelty: The concept of generalized m-derivations is newly introduced by us in ring theory in (1) and here we extend this concept to theory of semirings. We attempt to induce commutativity in weakly cancellative semirings (2) whose concept is unorthodox in the theory of semirings. This article pave new ways to study derivations and its applications on semirings.

Keywords: Derivations; generalized m-derivations; weakly cancellative semiring; commutativity