Original sentence: A homomorphism between two semigroups  and  is a function  such that.Any semigroup  may be embedded into a monoid (generally denoted as ) simply by adjoining an element  to  and defining  for all.Inverse semigroups
Edited sentence: A regular semigroup has plenty of idempotens of the form xy where.A semigroup with identity is called a monoid.A semigroup without identity may e embedded into a monoid simply by adjoining an element  to  and defining  for all.Alternatively, a semigroup is inverse if and only if any two idempotents commute.Analogous to the above construction, for every semigroup S, one defines S0, a semigroup with 0 that embeds S.Every element x has at least one inverse y satisfying xyx=x and yxy=x; x and y are sometimes called mutually inverse.Every semigroup, in fact every magma, has at most one identity element.Inverse semigroups are regular semigroups where every element has exactly one inverse.Similary, every magma has at most one absorbing element, which in semigroup theory is called a zero.The notation S1 denotes a monoid obtain from S by adjoining an identity if necessary (thus S1 = S for a monoid)