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dc.contributor.advisor Moss, Lawrence S en Aliyari, Saleh en 2010-12-13T21:03:08Z en 2027-08-13T20:03:08Z en 2012-02-10T02:52:05Z 2010-12-13T21:03:08Z en 2010 en
dc.identifier.uri en
dc.description Thesis (Ph.D.) - Indiana University, Mathematics, 2010 en
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 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. en
dc.language.iso EN en
dc.publisher [Bloomington, Ind.] : Indiana University en
dc.subject Countable en
dc.subject Modal en
dc.subject Normal en
dc.subject Ultrafilter extension en
dc.subject Conjecture en
dc.subject Canonical en
dc.subject Lindenbaum-Tarski Algebra en
dc.subject Bhirkhoff's Variety theorem en
dc.subject valuations en
dc.subject weak morphisms en
dc.subject Duality en
dc.subject Stone Vietoris en
dc.subject.classification Computer Science en
dc.subject.classification Logic en
dc.subject.classification Mathematics en
dc.title Topological Representation of Canonicity for Varieties of Modal Algebras en
dc.type Doctoral Dissertation en

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