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Title: Canonical extensions of bounded lattices and natural duality for default bilattices
Author: Craig, Andrew Philip Knott
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2012
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This thesis presents results concerning canonical extensions of bounded lattices and natural dualities for quasivarieties of default bilattices. Part I is dedicated to canonical extensions, while Part II focuses on natural duality for default bilattices. A canonical extension of a lattice-based algebra consists of a completion of the underlying lattice and extensions of the additional operations to the completion. Canonical extensions find rich application in providing an algebraic method for obtaining relational semantics for non-classical logics. Part I gives a new construction of the canonical extension of a bounded lattice. The construction is done via successive applications of functors and thus provides an elegant exposition of the fact that the canonical extension is functorial. Many existing constructions are described via representation and duality theorems. We demonstrate precisely how our new formulation relates to existing constructions as well as proving new results about complete lattices constructed from graphs. Part I ends with an analysis of the untopologised structures used in two methods of construction of canonical extensions of bounded lattices: the untopologised graphs used in our new construction, and the so-called `intermediate structure'. We provide sufficient conditions for the intermediate structure to be a lattice and, for the case of finite lattices, identify when the dual graph is not a minimal representation of the lattice. Part II applies techniques from natural duality theory to obtain dualities for quasivarieties of bilattices used in default logic. Bilattices are doubly-ordered algebraic structures which find application in reasoning about inconsistent and incomplete information. This account is the first attempt to provide dualities or representations when there is little interaction required between the two orders. Our investigations begin by using computer programs to calculate dualities for specific examples, before using purely theoretical techniques to obtain dualities for more general cases. The results obtained are extremely revealing, demonstrating how one of the lattice orders from the original algebra is encoded in the dual structure. We conclude Part II by describing a new class of default bilattices. These provide an alternative way of interpreting contradictory information. We obtain dualities for two newly-described quasivarieties and provide insights into how these dual structures relate to previously described classes of dual structures for bilattices.
Supervisor: Priestley, Hilary A. Sponsor: Rhodes Trust
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics ; canonical extension ; bounded lattice ; bilattice ; natural duality