  For an element a in a semigroup S the local subsemigroup of S with respect to
a is the subsemigroup aSa of S and the variant of S with respect to a is a
semigroup with underlying set S with a sandwich operation xy = xay for all x, y
in S. In this paper we discuss the structures of local subsemigroups of full
transformation semigroups and symmetric inverse monoids. It is also shown that
the set of all local subsemigroups of finite symmetric inverse monoids and the
set of all variants of all finite symmetric inverse monoids are same upto
isomorphism.

Group Theory