Proposition 1

Let M be a monoid operating on a set E and A a subset of E. (a) The stabilizer, strict stabilizer and fixer of A are submonoids of M. (b) Let $\alpha$ be an invertible element of M; if $\alpha$ belongs to the strict stabilizer (resp. fixer) of A, so does $\alpha^{-1}$.

Let e be the identity element of M; then $ea = a$ for every element $a \in A$ and therefore e belongs to the fixer of A. Let $\alpha$ and $\beta$ be elements of E which stabilize A. Then $(\alpha \beta)A = \alpha(\beta A) \subset \alpha A \subset A$ and therefore the stabilizer of A is a submonoid of M. Similarly for the strict stabilizer and fixer of A, whence (a). If $\alpha A = A$, then $A = \alpha^{-1}(\alpha A) = \alpha^{-1}A$. If for all $a \in A$, $\alpha a = a$, then $a = \alpha^{-1}(\alpha a) = \alpha^{-1}a$, whence (b).