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A categorical approach to lattice-like structures
Thesis (PhD)--Stellenbosch University, 2018.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2018
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| _version_ | 1867613800297922560 |
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| access_status_str | Open Access |
| author | Hoefnagel, Michael Anton |
| author2 | Janelidze, Zurab |
| author_browse | Hoefnagel, Michael Anton Janelidze, Zurab |
| author_facet | Janelidze, Zurab Hoefnagel, Michael Anton |
| author_sort | Hoefnagel, Michael Anton |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2018. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/104833 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:41:53.663Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2018 |
| publishDateRange | 2018 |
| publishDateSort | 2018 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/104833 A categorical approach to lattice-like structures Hoefnagel, Michael Anton Janelidze, Zurab Gray, James Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics. Categories (Mathematics) Mal'tsev categories Lattice (order) -- Mathematics Lattice structure UCTD Thesis (PhD)--Stellenbosch University, 2018. ENGLISH ABSTRACT : This thesis is a first step in a categorical approach to lattice-like structures. Its central notion, that of a majority category, relates to the category of lattices, in a similar way as Mal’tsev categories relate to the category of groups. This notion provides a context in which to establish categorical counterparts of various lattice-theoretic results. Surprisingly, many categories of a geometric nature naturally possess the dual property; namely, they are comajority categories. We show that several characterizations of varieties admitting a majority term, extend to characterizations of regular majority categories. These characterizations then show how majority categories relate to other well known notions in the literature, such as arithmetical and protoarithmetical categories. The most interesting results, from the point of view of the author, are those that concern decomposition and factorization. For example, every subobject of a finite product of objects in a regular majority category is uniquely determined by its two-fold projections – which can be seen as a certain subobject decomposition property. One of the main points of the thesis proves that in a regular majority category, every product of directly-irreducible objects is unique. AFRIKAANSE OPSOMMING : Hierdie proefskrif is ’n eerste stap na ’n kategoriese benadering tot roostersoos strukture. Die sentrale begrip daarvan, dié van ’n meerderheidskategorie, het betrekking op die kategorie van roosters, op soortgelyke wyse soos Mal’tsev-kategorieë betrekking het op die kategorie van groepe. Hierdie idee bied ’n konteks waarin kategoriese eweknieë van verskillende roosterteoretiese resultate gevestig kan word. Baie kategorieë van ’n meetkundige aard het die dubbele eienskap; naamlik, hulle is (co)meerderheids kategorieë. Ons wys dat verskeie karakters van variëteite wat ’n meerderheidstermyn toelaat, uitbrei na karakterisering van gereelde meerderheidskategorieë. Hierdie karakterisering toon dan aan hoe meerderheidskategorieë verband hou met ander bekende begrippe in die literatuur, soos Arithmetical en protoarithmetical kategorieë. Die mees interessante resultate, uit die oogpunt van die skrywer, is dié wat ontbinding en faktorisering betref. Ons wys dat direkte produkte erken ’n sekere unieke faktorisering stelling soortgelyk aan die universele algebraïese teendeel. Doctoral 2018-11-21T06:28:19Z 2018-12-07T06:47:04Z 2018-11-21T06:28:19Z 2018-12-07T06:47:04Z 2018-12 Thesis http://hdl.handle.net/10019.1/104833 en_ZA Stellenbosch University vii, 91 pages : illustrations application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Categories (Mathematics) Mal'tsev categories Lattice (order) -- Mathematics Lattice structure UCTD Hoefnagel, Michael Anton A categorical approach to lattice-like structures |
| title | A categorical approach to lattice-like structures |
| title_full | A categorical approach to lattice-like structures |
| title_fullStr | A categorical approach to lattice-like structures |
| title_full_unstemmed | A categorical approach to lattice-like structures |
| title_short | A categorical approach to lattice-like structures |
| title_sort | categorical approach to lattice like structures |
| topic | Categories (Mathematics) Mal'tsev categories Lattice (order) -- Mathematics Lattice structure UCTD |
| url | http://hdl.handle.net/10019.1/104833 |
| work_keys_str_mv | AT hoefnagelmichaelanton acategoricalapproachtolatticelikestructures AT hoefnagelmichaelanton categoricalapproachtolatticelikestructures |