On a reduction of full words and its applications to ideal elements in le-semigroups

It is known that the concept of [Formula: see text]-ideals in ordered semigroups generalizes several types of ideals. Full words play a crucial role in gaining a deeper understanding of [Formula: see text]-ideals in ordered semigroups. In this paper, we investigate the properties of full words using a relation called the reduction relation. Moreover, we extend the concept of [Formula: see text]-ideals in ordered semigroups to [Formula: see text]-ideal elements in [Formula: see text]-semigroups. Using results from the study of full word reductions, we establish connections among [Formula: see text]-ideal elements in [Formula: see text]-semigroups. Furthermore, we determine the generating forms of [Formula: see text]-ideal elements in [Formula: see text]-semigroups.