Universal theory of coalgebras

Tomasz Brengos

Abstract

The dissertation is devoted to the presentation of certain aspects of the universal the- ory of coalgebras. It is divided into three parts. The first part comprises an introduction to categories, functors and universal theory of coalgebras. It contains basic definitions and properties required in the thesis. The second part studies a dualization of a well-known Birkhoff-Malcev problem concerning a description of lattices of varieties of algebras. Here, we ask whether a full characterization of lattices of covarieties of coalgebras can be found. We prove that lattices of subcovarieties of a covariety К of T-coalgebras for a bounded functor T are exactly lattices isomorphic to the lattices of open subsets of topological spaces. Moreover, we show that covariety lattices for functors additionally preserving arbitrary non-empty intersections are precisely lattices isomorphic to the lattices of subcoalgebras of some PK-coalgebra. Finally, we narrow the class of Set-endofunctors we consider to finitary polynomial func- tors. In this case, we show that lattices isomorphic to lattices of subcoalgebras of some 'P
Diploma typeDoctor of Philosophy
Author Tomasz Brengos (FMIS / DAC)
Tomasz Brengos,,
- Department of Algebra and Combinatorics
Title in EnglishUniversal theory of coalgebras
Languageen angielski
Certifying UnitFaculty of Mathematics and Information Science (FMIS)
Disciplinemathematics / (mathematics domain) / (physical sciences)
Defense Date18-11-2010
Supervisor Anna Romanowska (FMIS / DAC)
Anna Romanowska,,
- Department of Algebra and Combinatorics

External reviewers Grzegorz Jarzembski - [Uniwersytet Mikołaja Kopernika w Toruniu]
Grzegorz Jarzembski,,
-
- Uniwersytet Mikołaja Kopernika w Toruniu

Denecke Klaus
Denecke Klaus,,
-
Pages86
Keywords in Englishxxx
Abstract in EnglishThe dissertation is devoted to the presentation of certain aspects of the universal the- ory of coalgebras. It is divided into three parts. The first part comprises an introduction to categories, functors and universal theory of coalgebras. It contains basic definitions and properties required in the thesis. The second part studies a dualization of a well-known Birkhoff-Malcev problem concerning a description of lattices of varieties of algebras. Here, we ask whether a full characterization of lattices of covarieties of coalgebras can be found. We prove that lattices of subcovarieties of a covariety К of T-coalgebras for a bounded functor T are exactly lattices isomorphic to the lattices of open subsets of topological spaces. Moreover, we show that covariety lattices for functors additionally preserving arbitrary non-empty intersections are precisely lattices isomorphic to the lattices of subcoalgebras of some PK-coalgebra. Finally, we narrow the class of Set-endofunctors we consider to finitary polynomial func- tors. In this case, we show that lattices isomorphic to lattices of subcoalgebras of some 'P
Thesis file
Brengos.pdf 601.82 KB

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