Vol. 34, No. 2, 1970

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Free topological semigroups and embedding topological semigroups in topological groups

Francis T. Christoph

Vol. 34 (1970), No. 2, 343–353

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.

Mathematical Subject Classification
Primary: 22.05
Received: 24 October 1969
Revised: 16 February 1970
Published: 1 August 1970
Francis T. Christoph