Ultraproducts and Los’s Theorem: A Category-Theoretic Analysis
| dc.contributor.advisor | Boxall, Gareth John | en_ZA |
| dc.contributor.author | Chimes, Mark Jonathan | en_ZA |
| dc.contributor.other | Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences | en_ZA |
| dc.date.accessioned | 2017-02-21T06:38:31Z | |
| dc.date.accessioned | 2017-03-29T12:19:59Z | |
| dc.date.available | 2017-02-21T06:38:31Z | |
| dc.date.available | 2017-03-29T12:19:59Z | |
| dc.date.issued | 2017-03 | |
| dc.description | Thesis (MSc)--Stellenbosch University, 2017 | en_ZA |
| dc.description.abstract | ENGLISH ABSTRACT : Ultraproducts are an important construction in model theory, especially as applied to algebra. Given some family of structures of a certain type, an ultraproduct of this family is a single structure which, in some sense, captures the important aspects of the family, where “important” is defined relative to a set of sets called an ultrafilter, which encodes which subfamilies are considered “large”. This follows from Lo´s’s Theorem, namely, the Fundamental Theorem of Ultraproducts, which states that every first-order sentence is true of the ultraproduct if, and only if, there is some “large” subfamily of the family such that it is true of every structure in this subfamily. In this dissertation, ultraproducts are examined both from the standard model-theoretic, as well as from the category-theoretic view. Some potential problems with the categorytheoretic definition of ultraproducts are pointed out, and it is argued that these are not as great an issue as first perceived. A general version of Lo´s’s Theorem is shown to hold for category-theoretic ultraproducts in general. This makes use of the concept of injectivity of a (compact) tree, which is intended to generalize truth of first-order formulae (under given assignments of variables), and, in the category of relational structures, corresponds exactly to first-order formulae. This type of thinking leads to a means of characterizing fields in the category of rings, and a new proof that every ultraproduct of fields is a field, which takes place entirely in the category of rings (along with the inclusion of the category of fields). Finally, the family of all (category-theoretic) ultraproducts on a given family is shown to arise from the “codensity monad" of the functor which includes the category of finite families into the category of families. In this sense, it is shown that ultraproducts are a rather natural construction category-theoretically speaking. | en_ZA |
| dc.description.abstract | AFRIKAANSE OPSOMMING : Ultraprodukte is ’n belangrike konstruksie in modelteorie, veral in hul toepassings in algebra. Gegewe ’n reeks strukture van ’n sekere tipe, is ’n ultraproduk van hierdie reeks ’n enkele struktuur wat, op ’n manier, die belangrikste aspekte van die reeks bevat, waar “belangrik” hier gedefinie¨er word met betrekking tot ’n versameling reekse wat ’n ultrafilter genoem word. Hierdie ultrafilter verteenwoordig watter subreekse deur die ultraproduk as “groot” beskou word. Dit is ’n gevolgtrekking van Lo´s se Stelling, dit wil sˆe, ’n eersteorde stelling is waar met betrekking tot die ultraproduk as, en slegs as, daar ’n “groot” subreeks (van die hoofreeks) bestaan sodat die stelling waar is met betrekking tot elke struktuur in di´e subreeks. In hierdie tesis word ultraprodukte uit die standarde model-teoretiese oogpunt behandel, sowel as uit die oogpunt van kategorie teorie. Potentie¨ele probleme met die kategorie-teoretiese ultraproduk word uitgelig, maar dit word geargumenteer dat hul nie so ’n groot probleem veroorsaak as wat dit blyk nie. ’n Algmene weergawe van Lo´s se stelling is bewys vir alle kategorie¨e. D´ıt maak gebruik van die konsep van injektiwiteit van ’n (kompakte) boom. Die bedoeling hiervan is om die waarheid van ’n eerste-orde stelling (onder ’n gegewe toedeling van veranderlikes) te veralgemeen. Hierdie idee ly tot ’n metode om liggame in die kategorie van groepe uit te lig, sowel as ’n nuwe bewys dat elke ultraproduk van liggame weer self ’n liggaam is. Hierdie bewys neem heeltemaal in die kategorie van groepe plaas (tesame met die funktor wat die kategorie van liggame in die kategorie van groepe insluit). Laastens, word dit angevoer dat die reeks van alle (kategorie-teoretiese) ultraprodukte van ’n gegewe reeks bestaan uit die “codigtheids monade” van die funktor wat die kategorie van eindige reekse insluit in die kategorie van oneindige reekse. Hierdie is dan ’n oortuiging dat ultraprodukte redelik natuurlik bestaan, ten minste uit die oogpunt van kategorie-teorie. | af_ZA |
| dc.format.extent | x, 183 pages | en_ZA |
| dc.identifier.uri | http://hdl.handle.net/10019.1/101202 | |
| dc.language.iso | en_ZA | en_ZA |
| dc.publisher | Stellenbosch : Stellenbosch University | en_ZA |
| dc.rights.holder | Stellenbosch University | en_ZA |
| dc.subject | Algebra, Abstract | en_ZA |
| dc.subject | Ultraproduct (Mathematics) | en_ZA |
| dc.subject | Category theory (Mathematics) | en_ZA |
| dc.subject | Model theory | en_ZA |
| dc.subject | Codensity | en_ZA |
| dc.subject | Categorical logic (Mathematics) | en_ZA |
| dc.subject | Ultrafilter (Mathematics) | en_ZA |
| dc.subject | Los's theorem (Mathematics) | en_ZA |
| dc.subject | Los's lemma (Mathematics) | en_ZA |
| dc.subject | Algebraic logic | en_ZA |
| dc.subject | UCTD | en_ZA |
| dc.title | Ultraproducts and Los’s Theorem: A Category-Theoretic Analysis | en_ZA |
| dc.type | Thesis | en_ZA |