# Indiana University

 dc.contributor.advisor Moss, Lawrence S en dc.contributor.author Aliyari, Saleh en dc.date.accessioned 2010-12-13T21:03:08Z en dc.date.available 2027-08-13T20:03:08Z en dc.date.available 2012-02-10T02:52:05Z dc.date.issued 2010-12-13T21:03:08Z en dc.date.submitted 2010 en dc.identifier.uri http://hdl.handle.net/2022/9865 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 en 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 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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