Paper Title:Some Structures of Idempotent Commutative Semigroup

Abstract

The algebraic theory of semigroups was developed by A. H. Clifford and G. B. Preston [2] and it was extended by several authors like David. McLean [4]. The algebraic theory of commutative semigroup was studied and extended by various authors like M. A. Taiclin [13], A. P. Biryukov [1]. In this paper, we have defined some structures of Idempotent Commutative Semigroup. We have given a notion of left(right) normal, left(right) quasi-normal, regular, normal, left(right) semi-normal, left(right) semi-regular, rectangular, reduced in a Idempotent Commutative Semigroup S. We have proved various theorems like an Idempotent Commutative Semigroup S is left(right) normal if and only if left(right) quasi- normal; S is regular implies normal and vice versa. Further we also proved that S is left(right) semi- normal if and only if left(right) semi-regular; S is left(right) quasinormal implies left(right) semi-regular and vice versa. We also verified that S is left(right) quasi-normal if and only if left(right) semi-normal. Further it is proved that every left singular with rectangular semigroup is reduced.


Keywords:Semigroup, Identity, Idempotent elements, Commutativity, rectangular and reduced elements.