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Skew compact semigroups
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<p>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 T<sub>2</sub> semigroups extends to this wider class. We show:</p> <p>A continuous skew compact semigroup is a semigroup with skew compact topology τ, such that the semigroup operation is continuous τ<sup>2</sup>→ τ. Each of these contains a unique minimal ideal which is an upper set with respect to the specialization order.</p> <p>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:</p> <p>It is a compact Hausdorff group if either the operation is cancellative, or there is a unique idempotent and S<sup>2</sup> = S.</p> <p>Its topology arises from its subinvariant quasimetrics.</p> <p>Each *-closed ideal ≠ S is contained in a proper open ideal.</p>
Title: Skew compact semigroups
Description:
<p>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 T<sub>2</sub> semigroups extends to this wider class.
We show:</p> <p>A continuous skew compact semigroup is a semigroup with skew compact topology τ, such that the semigroup operation is continuous τ<sup>2</sup>→ τ.
Each of these contains a unique minimal ideal which is an upper set with respect to the specialization order.
</p> <p>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:</p> <p>It is a compact Hausdorff group if either the operation is cancellative, or there is a unique idempotent and S<sup>2</sup> = S.
</p> <p>Its topology arises from its subinvariant quasimetrics.
</p> <p>Each *-closed ideal ≠ S is contained in a proper open ideal.
</p>.
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