Algebraic methods for hybrid logics

**Authors:**Robinson, Claudette**Date:**2015-07-02**Subjects:**Modality (Logic) , Hybrid logics , Algebra**Type:**Thesis**Identifier:**uj:13675 , http://hdl.handle.net/10210/13869**Description:**Ph.D. (Mathematics) , Algebraic methods have been largely ignored within the eld of hybrid logics. A main theme of this thesis is to illustrate the usefulness of algebraic methods in this eld. It is a well-known fact that certain properties of a logic correspond to properties of particular classes of algebras, and that we therefore can use these classes of algebras to answer questions about the logic. The rst aim of this thesis is to identify a class of algebras corresponding to hybrid logics. In particular, we introduce hybrid algebras as algebraic semantics for the better known hybrid languages in the literature. The second aim of this thesis is to use hybrid algebras to solve logical problems in the eld of hybrid logic. Specically, we will focus on proving general completeness results for some well-known hybrid logics with respect to hybrid algebras. Next, we study Sahlqvist theory for hybrid logics. We introduce syntactically de ned classes of hybrid formulas that have rst-order frame correspondents, which are preserved under taking Dedekind MacNeille completions of atomic hybrid algebras, and which are preserved under canonical extensions of permeated hybrid algebras. Finally, we investigate the nite model property (FMP) for several hybrid logics. In particular, we give analogues of Bull's theorem for the hybrid logics under consideration in this thesis. We also show that if certain syntactically de ned classes of hybrid formulas are added to the normal modal logic S4 as axioms, we obtain hybrid logics with the nite model property.**Full Text:**

**Authors:**Robinson, Claudette**Date:**2015-07-02**Subjects:**Modality (Logic) , Hybrid logics , Algebra**Type:**Thesis**Identifier:**uj:13675 , http://hdl.handle.net/10210/13869**Description:**Ph.D. (Mathematics) , Algebraic methods have been largely ignored within the eld of hybrid logics. A main theme of this thesis is to illustrate the usefulness of algebraic methods in this eld. It is a well-known fact that certain properties of a logic correspond to properties of particular classes of algebras, and that we therefore can use these classes of algebras to answer questions about the logic. The rst aim of this thesis is to identify a class of algebras corresponding to hybrid logics. In particular, we introduce hybrid algebras as algebraic semantics for the better known hybrid languages in the literature. The second aim of this thesis is to use hybrid algebras to solve logical problems in the eld of hybrid logic. Specically, we will focus on proving general completeness results for some well-known hybrid logics with respect to hybrid algebras. Next, we study Sahlqvist theory for hybrid logics. We introduce syntactically de ned classes of hybrid formulas that have rst-order frame correspondents, which are preserved under taking Dedekind MacNeille completions of atomic hybrid algebras, and which are preserved under canonical extensions of permeated hybrid algebras. Finally, we investigate the nite model property (FMP) for several hybrid logics. In particular, we give analogues of Bull's theorem for the hybrid logics under consideration in this thesis. We also show that if certain syntactically de ned classes of hybrid formulas are added to the normal modal logic S4 as axioms, we obtain hybrid logics with the nite model property.**Full Text:**

Power structures and their applications

**Authors:**Brink, Chris**Date:**2014-02-10**Subjects:**Algebra , Algebra, Universal**Type:**Thesis**Identifier:**uj:3724 , http://hdl.handle.net/10210/9104**Description:**Ph.D. (Mathematics) , This thesis reports on an interdisciplinary research programme: an investigation of power structures, and their applications in various fields. A power construction is an attempt to lift whatever structure may exist between the elements of a set to subsets of that set. The notions of structure considered here are algebraic, relational and topological. It is shown how power constructions are useful in a number of contexts in Mathematics, in Logic, in Computer Science and in the Philosophy of Science. The thesis is therefore an exercise in what may be called lateral research, where the aim is to look horizontally across disciplinary boundaries, identify common basic concepts, and use these to fertilise each field with results from the others. This differs from the more common vertical research method, the two manifestations of which (in Mathematics, in particular) are specialisation and generalisation. To specialise means to narrow down the field of investigation, as with a group theorist studying specifically Abelian groups. Generalisation moves in the opposite direction - one may attempt, for example, to generalise a result first proved for Abelian groups to the case of arbitrary groups. But, whether narrowing down or opening up, in vertical research mode it is the concept alone which is under investigation - its own nature, rather than its relationships to other concepts. In particular, vertical research pays little attention to the occurrence and application of the concept under investigation in other disciplines. It is sad that 'research', in Mathematics, is often unthinkingly equated with 'vertical research'. This is detrimental to scholarship in at least two ways. One is the training of new scientists - more particularly, of new PhD's. It is ironic that though the requirement for a PhD is almost universally held to be 'original research', or 'a contribution to knowledge', few things are in fact more orthodox and conformist than a PhD thesis. Here I refer not just to presentation (uniformity of which may be beneficial), but to methodology: few PhD candidates would dare to prejudice their chances with unpredictable examiners by venturing outside the paradigm of vertical research. A second effect (which is also a cause) of equating 'research' with 'vertical research' is the allocation of research funding. Project proposals and grant applications must be evaluated; this is usually done by peer review, and it seems clear that referees' reports emanating from a smallish fraternity of specialists will have a more enthusiastic ring than...**Full Text:**

**Authors:**Brink, Chris**Date:**2014-02-10**Subjects:**Algebra , Algebra, Universal**Type:**Thesis**Identifier:**uj:3724 , http://hdl.handle.net/10210/9104**Description:**Ph.D. (Mathematics) , This thesis reports on an interdisciplinary research programme: an investigation of power structures, and their applications in various fields. A power construction is an attempt to lift whatever structure may exist between the elements of a set to subsets of that set. The notions of structure considered here are algebraic, relational and topological. It is shown how power constructions are useful in a number of contexts in Mathematics, in Logic, in Computer Science and in the Philosophy of Science. The thesis is therefore an exercise in what may be called lateral research, where the aim is to look horizontally across disciplinary boundaries, identify common basic concepts, and use these to fertilise each field with results from the others. This differs from the more common vertical research method, the two manifestations of which (in Mathematics, in particular) are specialisation and generalisation. To specialise means to narrow down the field of investigation, as with a group theorist studying specifically Abelian groups. Generalisation moves in the opposite direction - one may attempt, for example, to generalise a result first proved for Abelian groups to the case of arbitrary groups. But, whether narrowing down or opening up, in vertical research mode it is the concept alone which is under investigation - its own nature, rather than its relationships to other concepts. In particular, vertical research pays little attention to the occurrence and application of the concept under investigation in other disciplines. It is sad that 'research', in Mathematics, is often unthinkingly equated with 'vertical research'. This is detrimental to scholarship in at least two ways. One is the training of new scientists - more particularly, of new PhD's. It is ironic that though the requirement for a PhD is almost universally held to be 'original research', or 'a contribution to knowledge', few things are in fact more orthodox and conformist than a PhD thesis. Here I refer not just to presentation (uniformity of which may be beneficial), but to methodology: few PhD candidates would dare to prejudice their chances with unpredictable examiners by venturing outside the paradigm of vertical research. A second effect (which is also a cause) of equating 'research' with 'vertical research' is the allocation of research funding. Project proposals and grant applications must be evaluated; this is usually done by peer review, and it seems clear that referees' reports emanating from a smallish fraternity of specialists will have a more enthusiastic ring than...**Full Text:**

The quasi center of a banach algebra

**Authors:**Van Wyk, Ettiene**Date:**2014-02-17**Subjects:**Algebra , Mathematics**Type:**Thesis**Identifier:**uj:4086 , http://hdl.handle.net/10210/9434**Description:**M.Sc. (Mathematics) , Please refer to full text to view abstract**Full Text:**

**Authors:**Van Wyk, Ettiene**Date:**2014-02-17**Subjects:**Algebra , Mathematics**Type:**Thesis**Identifier:**uj:4086 , http://hdl.handle.net/10210/9434**Description:**M.Sc. (Mathematics) , Please refer to full text to view abstract**Full Text:**

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