Definition 3

Let M be a monoid operating on a set E and A and B subsets of E. The set of $\alpha \in M$ such that $\alpha A \subset B$ (resp. $\alpha A = B$) is called the transporter (resp. strict transporter) of A to B. The transporter (resp. strict transporter) of A to A is called the stabilizer (resp. strict stabilizer) of A. The set of $\alpha \in M$ such that $\alpha a = a$ for all $a \in A$ is called the fixer of A.

An element $\alpha$ of M is said to stabilize (resp. stabilize strictly, resp. fix) a subset A of E if $\alpha$ belongs to the stabilizer (resp. strict stabilizer, resp. fixer) of A. A subset P of M is said to stabilize (resp. stabilize strictly, resp. fix) a subset A of E if all the elements of P stabilize (resp. strictly stabilize, resp. fix) A. The fixer of A is contained in the strict stabilizer of A which itself is contained in the stabilizer of A.