Editing Semigroup action (section)

===Acts and transformations===

It is often convenient (for instance if there is more than one act under consideration) to use a letter, such as <math>T</math>, to denote the function
:<math> T\colon S\times X \to X</math>
defining the <math>S</math>-action and hence write <math>T(s, x)</math> in place of <math>s\cdot x</math>. Then for any <math>s</math> in <math>S</math>, we denote by
:<math> T_s\colon X \to X</math>
the transformation of <math>X</math> defined by
:<math> T_s(x) = T(s,x).</math>
By the defining property of an <math>S</math>-act, <math>T</math> satisfies
:<math> T_{s*t} = T_s\circ T_t.</math>
Further, consider a function <math>s\mapsto T_s</math>. It is the same as <math>\operatorname{curry}(T):S\to(X\to X)</math> (see ''[[Currying]]''). Because <math>\operatorname{curry}</math> is a bijection, semigroup actions can be defined as functions <math>S\to(X\to X)</math> which satisfy
:<math> \operatorname{curry}(T)(s*t) = \operatorname{curry}(T)(s)\circ \operatorname{curry}(T)(t).</math>
That is, <math>T</math> is a semigroup action of <math>S</math> on <math>X</math> if and only if <math>\operatorname{curry}(T)</math> is a [[semigroup homomorphism]] from <math>S</math> to the full transformation monoid of <math>X</math>.