Abstract This paper introduces the notion of relatively commutative ordered semigroups as a generalization the commutativity definitions in ordered semigroups. Within this framework, the behaviour of products of two principal ideals generated by arbitrary subsets is considered, and it is proven that the product of the generated ideals is contained in a specific subset. Counterexamples establish that the reverse containment is generally not satisfied. Furthermore, a necessary and sufficient condition is established for such a semigroup to possess the structure of an inverse ordered semigroup. Utilizing the generalized definition of commutativity, it is proven that elements commute with Green’s J J -classes and H H -classes, and some examples of them are presented. Additionally, conditions are identified under which the Green’s L L -, R R -, J J -, and H H -classes coincide. Finally, the role of the radical of a subset is characterized in a relatively commutative ordered semigroup, revealing its significance in understanding the underlying algebraic properties of the ordered semigroup.
Mekera et al. (Wed,) studied this question.