Dual attachment pairs in categorically-algebraic topology
The paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category theory and universal algebra, and which is called categorically-algebraic (catalg) topology. The new framework is used to build a topological setting, based in a...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Universitat Politècnica de València
2011-10-01
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| Series: | Applied General Topology |
| Subjects: | |
| Online Access: | http://polipapers.upv.es/index.php/AGT/article/view/1646 |
| _version_ | 1819268360377991168 |
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| author | Anna Frascella Cosimo Guido Sergey A. Solovyov |
| author_facet | Anna Frascella Cosimo Guido Sergey A. Solovyov |
| author_sort | Anna Frascella |
| collection | DOAJ |
| description | The paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category theory and universal algebra, and which is called categorically-algebraic (catalg) topology. The new framework is used to build a topological setting, based in a catalg extension of the set-theoretic membership relation "e" called dual attachment, thereby dualizing the notion of attachment introduced by the authors earlier. Following the recent interest of the fuzzy community in topological systems of S. Vickers, we clarify completely relationships between these structures and (dual) attachment, showing that unlike the former, the latter have no inherent topology, but are capable of providing a natural transformation between two topological theories. We also outline a more general setting for developing the attachment theory, motivated by the concept of (L,M)-fuzzy topological space of T. Kubiak and A. Sostak. |
| first_indexed | 2024-12-23T21:31:49Z |
| format | Article |
| id | doaj.art-a630b34fb30b4e87af09f80269723fe3 |
| institution | Directory Open Access Journal |
| issn | 1576-9402 1989-4147 |
| language | English |
| last_indexed | 2024-12-23T21:31:49Z |
| publishDate | 2011-10-01 |
| publisher | Universitat Politècnica de València |
| record_format | Article |
| series | Applied General Topology |
| spelling | doaj.art-a630b34fb30b4e87af09f80269723fe32022-12-21T17:30:26ZengUniversitat Politècnica de ValènciaApplied General Topology1576-94021989-41472011-10-0112210113410.4995/agt.2011.16461348Dual attachment pairs in categorically-algebraic topologyAnna Frascella0Cosimo Guido1Sergey A. Solovyov2University of SalentoUniversity of SalentoUniversity of LatviaThe paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category theory and universal algebra, and which is called categorically-algebraic (catalg) topology. The new framework is used to build a topological setting, based in a catalg extension of the set-theoretic membership relation "e" called dual attachment, thereby dualizing the notion of attachment introduced by the authors earlier. Following the recent interest of the fuzzy community in topological systems of S. Vickers, we clarify completely relationships between these structures and (dual) attachment, showing that unlike the former, the latter have no inherent topology, but are capable of providing a natural transformation between two topological theories. We also outline a more general setting for developing the attachment theory, motivated by the concept of (L,M)-fuzzy topological space of T. Kubiak and A. Sostak.http://polipapers.upv.es/index.php/AGT/article/view/1646Dual attachment pair(lattice-valued) categorically-algebraic topology(L,M)-fuzzy topology(localic) algebra(pre)image operatorQuasi-coincidence relationQuasi-frameSpatializationTopological systemVariety |
| spellingShingle | Anna Frascella Cosimo Guido Sergey A. Solovyov Dual attachment pairs in categorically-algebraic topology Applied General Topology Dual attachment pair (lattice-valued) categorically-algebraic topology (L,M)-fuzzy topology (localic) algebra (pre)image operator Quasi-coincidence relation Quasi-frame Spatialization Topological system Variety |
| title | Dual attachment pairs in categorically-algebraic topology |
| title_full | Dual attachment pairs in categorically-algebraic topology |
| title_fullStr | Dual attachment pairs in categorically-algebraic topology |
| title_full_unstemmed | Dual attachment pairs in categorically-algebraic topology |
| title_short | Dual attachment pairs in categorically-algebraic topology |
| title_sort | dual attachment pairs in categorically algebraic topology |
| topic | Dual attachment pair (lattice-valued) categorically-algebraic topology (L,M)-fuzzy topology (localic) algebra (pre)image operator Quasi-coincidence relation Quasi-frame Spatialization Topological system Variety |
| url | http://polipapers.upv.es/index.php/AGT/article/view/1646 |
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