On Characterization of Relatively Commutative Semigroups

This paper defines a novel semigroup construction, called a relatively commutative semigroup. A semigroup is called relatively commutative if there exist $k,r,t,s\in \mathbb{Z^{+}}$ and $x,y\in S$ such that $(ab)^{k}=b^{r}ax$ and $(ab)^{t}=yba^{s}$ for all $a,b\in S$. The manuscript also provides significant results and examples exploring relatively commutative semigroups' structural characteristics. Additionally, the study addressing the relationships between various commutative semigroups and the relatively commutative semigroup demonstrates that every quasicommutative, $\mathcal{H}$-commutative, and $(m,n)$-commutative semigroup, as well as every normal semigroup and the center of a semigroup, is a relatively commutative semigroup. It further offers examples that illustrate that the reverse of the aforementioned relationships is not always applicable. Furthermore, it establishes that all semigroups with two elements are, in fact, either a left relatively commutative semigroup, a right relatively commutative semigroup, or a relatively commutative semigroup. Moreover, the paper confirms that while every commutative semigroup is also a relatively commutative semigroup, the reverse is valid when the relatively commutative semigroup has order 2. Besides, the paper delineates the necessary conditions for a one-sided ideal as an ideal within a relatively commutative semigroup, and demonstrates that if a relatively commutative semigroup is regular, it is reduced.