Skew compact semigroups

Ralph D. Kopperman, D. Robbie

Abstract

Skew compact spaces are the best behaving generalization of compact Hausdorff spaces to non-Hausdorff spaces. They are those (X ; τ ) such that there is another topology τ* on X for which τ V τ* is compact and (X; τ ; τ*) is pairwise Hausdorff; under these conditions, τ uniquely determines τ *, and (X; τ*) is also skew compact. Much of the theory of compact T2 semigroups extends to this wider class. We show:

A continuous skew compact semigroup is a semigroup with skew compact topology τ, such that the semigroup operation is continuous τ2→ τ. Each of these contains a unique minimal ideal which is an upper set with respect to the specialization order.

A skew compact semigroup which is a continuous semigroup with respect to both topologies is called a de Groot semigroup. Given one of these, we show:

It is a compact Hausdorff group if either the operation is cancellative, or there is a unique idempotent and S2 = S.

Its topology arises from its subinvariant quasimetrics.

Each *-closed ideal ≠ S is contained in a proper open ideal.


Keywords

Continuity space; de Groot (cocompact) dual; de Groot map; de Groot skew compact semigroup; Order-Hausdorff space; Skew compact space; Saturated set; Specialization order of a topology

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References

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Universitat Politècnica de València

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