Vol. 49, No. 2, 1973

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On Σ-inverse semigroups

S. Sribala

Vol. 49 (1973), No. 2, 483–488

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.

Mathematical Subject Classification 2000
Primary: 22A15
Secondary: 20M10
Received: 11 August 1972
Published: 1 December 1973
S. Sribala