Editing Semigroup (section)

=== Identity and zero ===
A '''[[left identity]]''' of a semigroup ''S'' (or more generally, [[magma (algebra)|magma]]) is an element ''e'' such that for all ''x'' in ''S'', {{nowrap|1=''e'' ⋅ ''x'' = ''x''}}.  Similarly, a '''[[right identity]]''' is an element ''f'' such that for all ''x'' in ''S'', {{nowrap|1=''x'' ⋅ ''f'' = ''x''}}. Left and right identities are both called '''one-sided identities'''.  A semigroup may have one or more left identities but no right identity, and vice versa.

A '''two-sided identity''' (or just '''identity''') is an element that is both a left and right identity.  Semigroups with a two-sided identity are called '''[[monoid]]s'''.  A semigroup may have at most one two-sided identity.  If a semigroup has a two-sided identity, then the two-sided identity is the only one-sided identity in the semigroup.  If a semigroup has both a left identity and a right identity, then it has a two-sided identity (which is therefore the unique one-sided identity).

A semigroup ''S'' without identity may be [[embedding|embedded]] in a monoid formed by adjoining an element {{nowrap|''e'' ∉ ''S''}} to ''S'' and defining {{nowrap|1=''e'' ⋅ ''s'' = ''s'' ⋅ ''e'' = ''s''}} for all {{nowrap|''s'' ∈ ''S'' ∪ {{mset|''e''}}}}.{{sfn|ps=|Jacobson|2009|p=30|loc=ex. 5}}{{sfn|ps=|Lawson|1998|p=[{{Google books|plainurl=y|id=_F78nQEACAAJ|page=20|text=adjoining an identity}} 20]}} The notation ''S''<sup>1</sup> denotes a monoid obtained from ''S'' by adjoining an identity ''if necessary'' ({{nowrap|1=''S''<sup>1</sup> = ''S''}} for a monoid).{{sfn|ps=|Lawson|1998|p=[{{Google books|plainurl=y|id=_F78nQEACAAJ|page=20|text=adjoining an identity}} 20]}}

Similarly, every magma has at most one [[absorbing element]], which in semigroup theory is called a '''zero'''. Analogous to the above construction, for every semigroup ''S'', one can define ''S''<sup>0</sup>, a semigroup with 0 that embeds ''S''.