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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.

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    Related (but not exactly the same that you asked): http://en.wikipedia.org/wiki/Center_(group_theory)2015-04-06

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