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# Topological Representation of Canonicity for Varieties of Modal Algebras

## DSpace/Manakin Repository

 dc.contributor.advisor Moss, Lawrence S en_US dc.contributor.author Aliyari, Saleh en_US dc.date.accessioned 2010-12-13T21:03:08Z dc.date.available 2027-08-13T20:03:08Z dc.date.available 2012-02-10T02:52:05Z dc.date.issued 2010-12-13T21:03:08Z dc.date.submitted 2010 en_US dc.identifier.uri http://hdl.handle.net/2022/9865 dc.description Thesis (Ph.D.) - Indiana University, Mathematics, 2010 en_US dc.description.abstract The main subject of this dissertation is to approach the question of countable canonicity of varieties of modal algebras from a topological and categorical point of view. The category of coalgebras of the Vietoris functor on the category of Stone spaces provides a class of frames we call sv-frames. We show that the semantic of this frames is equivalent to that of modal algebras so long as we are limited to certain valuations called sv-valuations. We show that the canonical frame of any normal modal logic en_US which is directly constructed based on the logic is an sv-frame. We then define the notion of canonicity of a logic in terms of varieties and their dual classes. We will then prove that any morphism on the category of coalgebras of the Vietoris functor whose codomain is the canonical frame of the minimal normal modal logic are exactly the ones that are invoked by sv-valuations. We will then proceed to reformulate canonicity of a variety of modal algebras determined by a logic in terms of properties of the class of sv-frames that correspond to that logic. We define ultrafilter extension as an operator on the category of sv-frames, prove a coproduct preservation result followed by some equivalent forms of canonicity. Using Stone duality the notion of co-variety of sv-frames is defined. The notion of validity of a logic on a frame is presented in terms of ranges of theory maps whose domain is the given frame. Partial equivalent results on co-varieties of sv-frames are proved. We classify theory maps which are maps invoked by a valuation on a Kripke frame using the classification of sv-theory maps and properties of ultrafilter extension. A negative categorical result concerning the existence of an adjoint functor for ultrafilter extension is also proved. dc.language.iso EN en_US dc.publisher [Bloomington, Ind.] : Indiana University en_US dc.subject Countable en_US dc.subject Modal en_US dc.subject Normal en_US dc.subject Ultrafilter extension en_US dc.subject Conjecture en_US dc.subject Canonical en_US dc.subject Lindenbaum-Tarski Algebra dc.subject Bhirkhoff's Variety theorem dc.subject valuations dc.subject weak morphisms dc.subject Duality dc.subject Stone Vietoris dc.subject.classification Computer Science en_US dc.subject.classification Logic en_US dc.subject.classification Mathematics en_US dc.title Topological Representation of Canonicity for Varieties of Modal Algebras en_US dc.type Doctoral Dissertation en_US
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