In this paper, the
Preston-Vagner theorem on representation of inverse semigroups is extended to a
class of uniform inverse semigroups. In this connection the notion of Σ-uniformity on
an inverse simigroup is introduced which is a modification of the congruence
uniformity defined by a set of idempotent separating congruences on the
inverse semigroup. Such an inverse semigroup is called a Σ-inverse semigroup.
First, it is proved that a Σ-inverse semigroup is complete if and only if all
its maximal subgroups are complete and it is compact if and only if the
set of its idempotents is finite and all its maximal subgroups are compact.
Next, the symmetric Σ-inverse semigroup of bi-Lipchitzian maps between
U-open subsets of an uniform space is defined and finally, it is shown that any
Σ-inverse semigroup can be embedded isomorphically into a symmetric Σ-inverse
semigroup.
