Let $G$ be a group and $S$ its subset. I would like to consider the following condition on $S$.
For every $x,y\in S,$ we have $xy=yx.$
This is trivially equivalent to $S\subseteq C(S).$
The same condition can be formulated for a semigroup, and if we define the centralizer of a subset of a semigroup in the same way as for a group, then the equivalence still obviously holds.
I would like to know if there is a name for this condition.