Upper and Lower Periodic Subsets of Semigroups

Abstract

In this paper, a new topic about a vast class of subsets of semigroups and binary systems, which contains all ideals, periodic subsets and sub-semigroups, is introduced and studied. In fact, the “upper periodic subsets” can be considered as a generalization of the conception “ideals”. We prove a fundamental theorem which states that if A is a (left) upper B-periodic subset of a semigroup S, then under some conditions, it has a unique direct representation [Formula: see text], where B1=B ∪ {1} and B ⊆ 𝔅 ≤ S. Especially, we prove a unique direct representation for upper and lower T-periodic subsets, and classify all sub-semigroups of S containing a fixed element T to three classes. This classification gives us more interesting properties for the real semigroups. At last, we characterize upper and lower T-periodic subsets of semigroups and groups.