Editing Semigroup action (section)

==Formal definitions==

Let ''S'' be a semigroup. Then a (left) '''semigroup action''' (or '''act''') of ''S'' is a set ''X'' together with an operation {{nowrap|• : ''S'' × ''X'' → ''X''}} which is compatible with the semigroup [[binary operation|operation]] ∗ as follows:
* for all ''s'', ''t'' in ''S'' and ''x'' in ''X'', {{nowrap|1=''s'' • (''t'' • ''x'') = (''s'' ∗ ''t'') • ''x''}}.
This is the analogue in semigroup theory of a (left) [[Group action (mathematics)|group action]], and is equivalent to a [[semigroup#Identity and zero|semigroup homomorphism]] into the set of functions on ''X''. Right semigroup actions are defined in a similar way using an operation {{nowrap|• : ''X'' × ''S'' → ''X''}} satisfying {{nowrap|1=(''x'' • ''a'') • ''b'' = ''x'' • (''a'' ∗ ''b'')}}.

If ''M'' is a monoid, then a (left) '''monoid action''' (or '''act''') of ''M'' is a (left) semigroup action of ''M'' with the additional property that
* for all ''x'' in ''X'': ''e'' • ''x'' = ''x''
where ''e'' is the identity element of ''M''. This correspondingly gives a monoid homomorphism. Right monoid actions are defined in a similar way. A monoid ''M'' with an action on a set is also called an '''operator monoid'''.

A semigroup action of ''S'' on ''X'' can be made into monoid act by adjoining an identity to the semigroup and requiring that it acts as the identity transformation on ''X''.

===Terminology and notation===

If ''S'' is a semigroup or monoid, then a set ''X'' on which ''S'' acts as above (on the left, say) is also known as a (left) '''''S''-act''', '''''S''-set''', '''''S''-action''', '''''S''-operand''', or '''left act over ''S'''''. Some authors do not distinguish between semigroup and monoid actions, by regarding the identity axiom ({{nowrap|1=''e'' • ''x'' = ''x''}}) as empty when there is no identity element, or by using the term '''unitary ''S''-act''' for an ''S''-act with an identity.<ref>Kilp, Knauer and Mikhalev, 2000, pages 43–44.</ref>

The defining property of an act is analogous to the [[associativity]] of the semigroup operation, and means that all parentheses can be omitted. It is common practice, especially in computer science, to omit the operations as well so that both the semigroup operation and the action are indicated by juxtaposition. In this way [[string (computer science)|strings]] of letters from ''S'' act on ''X'', as in the expression ''stx'' for ''s'', ''t'' in ''S'' and ''x'' in ''X''.

It is also quite common to work with right acts rather than left acts.<ref>Kilp, Knauer and Mikhalev, 2000.</ref> However, every right S-act can be interpreted as a left act over the '''opposite semigroup''', which has the same elements as S, but where multiplication is defined by reversing the factors, {{nowrap|1=''s'' • ''t'' = ''t'' • ''s''}}, so the two notions are essentially equivalent. Here we primarily adopt the point of view of left acts.

===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>.

===''S''-homomorphisms===

Let ''X'' and ''X''′ be ''S''-acts. Then an ''S''-homomorphism from ''X'' to ''X''′ is a map
:<math>F\colon X\to X'</math>
such that 
:<math>F(sx) =s F(x)</math> for all <math>s\in S</math> and <math>x\in X</math>.
The set of all such ''S''-homomorphisms is commonly written as <math>\mathrm{Hom}_S(X,X')</math>.

''M''-homomorphisms of ''M''-acts, for ''M'' a monoid, are defined in exactly the same way.

===''S''-Act and ''M''-Act===

For a fixed semigroup ''S'', the left ''S''-acts are the objects of a category, denoted ''S''-Act, whose morphisms are the ''S''-homomorphisms. The corresponding category of right ''S''-acts is sometimes denoted by Act-''S''. (This is analogous to the [[Category of modules|categories ''R''-Mod and Mod-''R'']] of left and right [[module (mathematics)|modules]] over a [[ring (mathematics)|ring]].)

For a monoid ''M'', the categories ''M''-Act and Act-''M'' are defined in the same way.