Preservation theorems for algebraic and relational models of logic
A thesis submitted to the School of Computer Science, Faculty of Science, University of the Witwatersrand, Johannesburg in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 15 May 2013 === In this thesis a number of different constructions on ordered algebraic s...
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ndltd-netd.ac.za-oai-union.ndltd.org-wits-oai-wiredspace.wits.ac.za-10539-129212019-05-11T03:41:32Z Preservation theorems for algebraic and relational models of logic Morton, Wilmari Ordered algebraic structures. Algebra. A thesis submitted to the School of Computer Science, Faculty of Science, University of the Witwatersrand, Johannesburg in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 15 May 2013 In this thesis a number of different constructions on ordered algebraic structures are studied. In particular, two types of constructions are considered: completions and finite embeddability property constructions. A main theme of this thesis is to determine, for each construction under consideration, whether or not a class of ordered algebraic structures is closed under the construction. Another main focus of this thesis is, for a particular construction, to give a syntactical description of properties preserved by the construction. A property is said to be preserved by a construction if, whenever an ordered algebraic structure satisfies it, then the structure obtained through the construction also satisfies the property. The first four constructions investigated in this thesis are types of completions. A completion of an ordered algebraic structure consists of a completely lattice ordered algebraic structure and an embedding that embeds the former into the latter. Firstly, different types of filters (dually, ideals) of partially ordered sets are investigated. These are then used to form the filter (dually, ideal) completions of partially ordered sets. The other completions of ordered algebraic structures studied here include the MacNeille completion, the canonical extension (also called the completion with respect to a polarization) and finally a prime filter completion. A class of algebras has the finite embeddability property if every finite partial subalgebra of some algebra in the class can be embedded into some finite algebra in the class. Firstly, two constructions that establish the finite embeddability property for residuated ordered structures are investigated. Both of these constructions are based on completion constructions: the first on the Mac- Neille completion and the second on the canonical extension. Finally, algebraic filtrations on modal algebras are considered and a duality between algebraic and relational versions of filtrations is established. 2013-07-30T12:03:32Z 2013-07-30T12:03:32Z 2013-07-30 Thesis http://hdl.handle.net/10539/12921 en application/pdf |
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Ordered algebraic structures. Algebra. |
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Ordered algebraic structures. Algebra. Morton, Wilmari Preservation theorems for algebraic and relational models of logic |
description |
A thesis submitted to the School of Computer Science,
Faculty of Science,
University of the Witwatersrand, Johannesburg
in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 15 May 2013 === In this thesis a number of different constructions on ordered algebraic structures
are studied. In particular, two types of constructions are considered: completions
and finite embeddability property constructions.
A main theme of this thesis is to determine, for each construction under
consideration, whether or not a class of ordered algebraic structures is closed
under the construction. Another main focus of this thesis is, for a particular
construction, to give a syntactical description of properties preserved by the
construction. A property is said to be preserved by a construction if, whenever
an ordered algebraic structure satisfies it, then the structure obtained through
the construction also satisfies the property.
The first four constructions investigated in this thesis are types of completions.
A completion of an ordered algebraic structure consists of a completely
lattice ordered algebraic structure and an embedding that embeds the former
into the latter. Firstly, different types of filters (dually, ideals) of partially ordered
sets are investigated. These are then used to form the filter (dually, ideal)
completions of partially ordered sets. The other completions of ordered algebraic
structures studied here include the MacNeille completion, the canonical
extension (also called the completion with respect to a polarization) and finally
a prime filter completion.
A class of algebras has the finite embeddability property if every finite partial
subalgebra of some algebra in the class can be embedded into some finite
algebra in the class. Firstly, two constructions that establish the finite embeddability
property for residuated ordered structures are investigated. Both of
these constructions are based on completion constructions: the first on the Mac-
Neille completion and the second on the canonical extension. Finally, algebraic
filtrations on modal algebras are considered and a duality between algebraic and
relational versions of filtrations is established. |
author |
Morton, Wilmari |
author_facet |
Morton, Wilmari |
author_sort |
Morton, Wilmari |
title |
Preservation theorems for algebraic and relational models of logic |
title_short |
Preservation theorems for algebraic and relational models of logic |
title_full |
Preservation theorems for algebraic and relational models of logic |
title_fullStr |
Preservation theorems for algebraic and relational models of logic |
title_full_unstemmed |
Preservation theorems for algebraic and relational models of logic |
title_sort |
preservation theorems for algebraic and relational models of logic |
publishDate |
2013 |
url |
http://hdl.handle.net/10539/12921 |
work_keys_str_mv |
AT mortonwilmari preservationtheoremsforalgebraicandrelationalmodelsoflogic |
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1719083764859011072 |