Transformation

A mapping $u$ of a set $X$ (in general endowed with some structure) into itself. The image of an element $x \in X$ under the transformation $u$ is denoted by $u(x)$, $ux$, $x u$ or $x^u$. The set of all transformations of a set $X$ into itself forms a monoid with respect to multiplication (composition), with the identity map as identity element, which is called the symmetric transformation semi-group on $X$. The invertible elements of this semi-group are called permutations (cf. Permutation of a set). All permutations on a set $X$ form a subgroup of the symmetric semi-group — the symmetric group on $X$.