The aim of writing this paper is given in the title. We want to show that not only the ideals but also the ideal elements play an essential role in studying the structure of some ordered semigroups.We first prove that a $\vee e$-semigroup $S$ is a semilattice of left simple $\vee e$-semigroups if and only if it is decomposable into some pairwise disjoint left simple $\vee e$-subsemigroups of $S$ indexed by a semilattice $Y$. Then we give an example of a semilattice of left simple $\vee e$-semigroups that leads to a characterization of the semilattices of left simple and the chains of left simple $\vee e$-semigroups in terms of left ideal elements.
"The role of the ideal elements in studying the structure of some ordered semigroups,"
Turkish Journal of Mathematics: Vol. 41:
5, Article 6.
Available at: https://journals.tubitak.gov.tr/math/vol41/iss5/6