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 wellknown BirkhoffMalcev 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 Tcoalgebras 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 nonempty intersections are precisely lattices isomorphic to the lattices of subcoalgebras of some PKcoalgebra. Finally, we narrow the class of Setendofunctors we consider to finitary polynomial func tors. In this case, we show that lattices isomorphic to lattices of subcoalgebras of some 'PDiploma type  Doctor of Philosophy  
Author 
Tomasz Brengos (FMIS / DAC)
Tomasz Brengos
 
Title in English  Universal theory of coalgebras  
Language  en angielski  
Certifying Unit  Faculty of Mathematics and Information Science (FMIS)  
Discipline  mathematics / (mathematics domain) / (physical sciences)  
Defense Date  18112010  
Supervisor 
Anna Romanowska (FMIS / DAC)
Anna Romanowska
 
External reviewers 
Grzegorz Jarzembski  [Uniwersytet Mikołaja Kopernika w Toruniu]
Grzegorz Jarzembski
 Uniwersytet Mikołaja Kopernika w Toruniu Denecke Klaus
Denecke Klaus
 
Pages  86  
Keywords in English  xxx  
Abstract in English  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 wellknown BirkhoffMalcev 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 Tcoalgebras 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 nonempty intersections are precisely lattices isomorphic to the lattices of subcoalgebras of some PKcoalgebra. Finally, we narrow the class of Setendofunctors we consider to finitary polynomial func tors. In this case, we show that lattices isomorphic to lattices of subcoalgebras of some 'P  
Thesis file 

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