Simplicity in le-semigroups via partitions of full words, and its applications to regularity conditions induced by linear inequalities

Abstract The concept of simplicity in le -semigroups has been used to characterize the coincidence of ideal elements and to study Khayopulu–Green’s relations. Several types of simplicities are known, including two-sided simplicity, left simplicity, right simplicity, bisimplicity, quasisimplicity, and biinterior simplicity. A natural question arises: are there further notions of simplicity for le -semigroups? If so, how are they related? By employing the notion of partition ideal elements, we observe that several potential simplicities have not yet been explored. In this paper, we introduce new forms of simplicity in le -semigroups defined via partitions of full words, and we investigate the connections among these simplicities. Finally, we apply these notions of simplicity to determine the regularities and ideal elements of le -semigroups.