A characterization of principal ideals and Green’s relations in relatively commutative ordered semigroups

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">J</m:mi> </m:math> J -classes and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">H</m:mi> </m:math> H -classes, and some examples of them are presented. Additionally, conditions are identified under which the Green’s <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">L</m:mi> </m:math> L -, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">R</m:mi> </m:math> R -, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">J</m:mi> </m:math> J -, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">H</m:mi> </m:math> 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.