The purpose of this
paper is to study the problem of embedding a topological semigroup in a
topological group. A construction is given for a free topological semigroup
generated by a topological space. This construction is used to define a concept
called Property E and it is shown that a T1 completely regular topological
semigroup S can be embedded in a topological group if and only if S can be
embedded in a group and S has Property E. This generalizes a result of
Rothman who considered the problem of embedding a commutative, cancellative
topological semigroup in its group of quotients. Rothman’s results on embedding
certain subsemigroups of compact semigroups in topological groups are also
generalized.
