Editing Semigroup action (section)

===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.