Some generalizations of Lagrange theorem and factor subsets for semigroups

Abstract

It is well-known that  every group is equal to the direct product of  its subgroup and related left and right transversalsets (in the sense of direct product of subsets). Therefore,  every subgroup of a group is its left and right factor,and an its consequence is the Lagrange theorem for finite groups.This paper generalizes the results for semigroupsand proves a necessary and sufficient condition for a subgroup of asemigroup to be a factor.Also, by using the conception  upper periodic subsets of semigroups and groups (introduced byM.H. Hooshmand as a generalization of the conception ideals) we provesome sufficient conditions for a vast class of subsets of semigroups to be factorsand Lagrange subsets.