Some Generalizations of Lagrange Theorem and Factor Subsets for Semigroups

It is well known that every group is equal to the direct product of its subgroup and related left and right transversal sets (in the sense of direct product of subsets). therefore, every subgroup of a group is its left and right factor and one of its consequence is the lagrange’s theorem for finite groups. this paper generalizes the results for semigroups and proves a necessary and sufficient condition for a subgroup of a semigroup to be a factor. also, by using the conception upper periodic subsets of semigroups and groups (introduced by the author as a generalization of the conception ideals) we prove some sufficient conditions for a vast class of subsets of semigroups to be factors and lagrange subsets.