Skew compact semigroups
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...
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| Format: | Article |
| Language: | English |
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Universitat Politècnica de València
2003-04-01
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| Series: | Applied General Topology |
| Subjects: | |
| Online Access: | http://polipapers.upv.es/index.php/AGT/article/view/2015 |
| _version_ | 1811223130376503296 |
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| author | Ralph D. Kopperman Desmond Robbie |
| author_facet | Ralph D. Kopperman Desmond Robbie |
| author_sort | Ralph D. Kopperman |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-12T08:27:49Z |
| format | Article |
| id | doaj.art-ad46e848a1804984995b0a5e625c659d |
| institution | Directory Open Access Journal |
| issn | 1576-9402 1989-4147 |
| language | English |
| last_indexed | 2024-04-12T08:27:49Z |
| publishDate | 2003-04-01 |
| publisher | Universitat Politècnica de València |
| record_format | Article |
| series | Applied General Topology |
| spelling | doaj.art-ad46e848a1804984995b0a5e625c659d2022-12-22T03:40:19ZengUniversitat Politècnica de ValènciaApplied General Topology1576-94021989-41472003-04-014113314210.4995/agt.2003.20151636Skew compact semigroupsRalph D. Kopperman0Desmond Robbie1City University of New YorkUniversity of MelbourneSkew 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.http://polipapers.upv.es/index.php/AGT/article/view/2015Continuity spacede Groot (cocompact) dualde Groot mapde Groot skew compact semigroupOrder-Hausdorff spaceSkew compact spaceSaturated setSpecialization order of a topology |
| spellingShingle | Ralph D. Kopperman Desmond Robbie Skew compact semigroups Applied General Topology 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 |
| title | Skew compact semigroups |
| title_full | Skew compact semigroups |
| title_fullStr | Skew compact semigroups |
| title_full_unstemmed | Skew compact semigroups |
| title_short | Skew compact semigroups |
| title_sort | skew compact semigroups |
| topic | 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 |
| url | http://polipapers.upv.es/index.php/AGT/article/view/2015 |
| work_keys_str_mv | AT ralphdkopperman skewcompactsemigroups AT desmondrobbie skewcompactsemigroups |