On (m,n)-Anti-ideals in Ordered Semigroups

In this paper, we review the concept of (m,n)-anti-ideals in semigroups. This notion extends the classical concept of anti-ideals and is closely related to the study of mutants in semigroups. It is shown that a semigroup possessing a left identity admits no left (m,m)-anti-ideals, and that no (m,n)-anti-ideals exist when m < n. These preliminary results serve as a foundation for the subsequent development of the theory in the context of ordered semigroups.