Full Text Available
Note: Clicking the button above will open the full text document at the original institutional repository in a new window.
On reverse representation
Joubert, Gideo. 2024. On reverse representation. Unpublished masters dissertation. Stellenbosch : Stellenbosch University [online]. Available: https://scholar.sun.ac.za/handle/10019.1/131689 Thesis (MSc)--Stellenbosch University, 2024.
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | English |
| Published: |
Stellenbosch : Stellenbosch University
2025
|
| Subjects: | |
| Tags: |
No Tags, Be the first to tag this record!
|
| _version_ | 1867613774502952960 |
|---|---|
| access_status_str | Open Access |
| author | Joubert, Gideo |
| author2 | Faul, P. F. |
| author_browse | Faul, P. F. Joubert, Gideo |
| author_facet | Faul, P. F. Joubert, Gideo |
| author_sort | Joubert, Gideo |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Joubert, Gideo. 2024. On reverse representation. Unpublished masters dissertation. Stellenbosch : Stellenbosch University [online]. Available: https://scholar.sun.ac.za/handle/10019.1/131689
Thesis (MSc)--Stellenbosch University, 2024. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/131689 |
| institution | Stellenbosch University (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:41:29.531Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2025 |
| publishDateRange | 2025 |
| publishDateSort | 2025 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/131689 On reverse representation Joubert, Gideo Faul, P. F. Janelidze, Zurab Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Cayley algebras Monoids Buildings (Group theory) Semigroups UCTD Joubert, Gideo. 2024. On reverse representation. Unpublished masters dissertation. Stellenbosch : Stellenbosch University [online]. Available: https://scholar.sun.ac.za/handle/10019.1/131689 Thesis (MSc)--Stellenbosch University, 2024. ENGLISH ABSTRACT: Arthur Cayley’s famous representation theorem demonstrates to us that every group G can be realized as a subgroup of the group of automorphisms over some set X. Building on this foundational theorem, one can easily prove similar results for monoids and most generally faithful semigroups. The class of all semigroups with the property that ∀x,y∈X∀z∈X [xz = yz ⇒ x = y]. In this thesis, I establish the concept of reverse Cayley representation; a method of associating with any faithful semigroup S of transformations of a set X a (possibly empty) set of semigroups called unrepresentations. The set of unrepresentations of S is the set of exactly those faithful semigroups X on X with the property that X is represented as S. The goal of this thesis is to delve into reverse representation by defining and characterizing the unrepresentations of transformation structures at various levels of generality. We will look at a number of structures from the previously mentioned faithful semigroups of transformations to transformation groups. At each level I explore the fashion in which these unrepresentations arise and connect them to previously existing, or new, concepts. As I explore these unrepresentations, I will demonstrate that the set of unrepresentations of S is not merely a collection of structures, but has an algebraic structure itself. It is a heap, and this heap of unrepresentations is closely related to the group of ‘internal symmetries’ of any unrepresentation of S. This insight enables us to extend the concept of an invertible element into the realm of semigroups. In addition to this foundational work, I also investigate two special cases of unrepresentation: In the context of Clifford semigroups, Is how h ow t he unrepresentation of a structure can be broken up into various ‘smaller’ unrepresentations of component structures (along with an condition which specifies how these smaller unrepresentations should interact). Furthermore, in the context of category theory, I illustrate how the concept of representation (and thus unrepresentation) can be generalized- not from a group to a semigroup, but from a set to a graph. AFRIKAANSE OPSOMMING: Arthur Cayley se beroemde voorstellingstelling demonstreer dat elke groep G as ’n ondergroep van die groep van automorfismes o or ’ n s tel X g erealiseer k an w ord. G ebaseer o p hierdie fundamentele resultaat, kan dit maklik veralgemeen word om monoïede en getroue semigroepe, die klas van alle semigroepe met die eienskap dat ∀ₓ,y∈X∀z∈X [xz = yz ⇒ x = y], op hierdie manier voor te stel. In hierdie tesis vestig ek die konsep van omgekeerde Cayley voorstelling; ’n metode om met enige getroue semigroep S van transformasies van ’n stel X ’n (moontlik leë) stel semigroepe te assosieer wat onverteenwoordigings genoem word. Die stel van onverteenwoordigings van S is die stel van presies daardie getroue semigroepe X op X met die eienskap dat X as S voorgestel word. Die doel van hierdie tesis is om omgekeerde voorstelling te verken deur die onverteen- woordigings van transformasiestrukture op verskeie vlakke van algemeneheid te definieer en te karakteriseer. Ons sal verskeie strukture van die voorheen genoemde getroue semigroepe van transformasies tot transformasiegroepe ondersoek. Op elke vlak sal ek die wyse waarop hier- die onverteenwoordigings ontstaan, verken en hulle verbind met voorheen bestaande of nuwe konsepte. Terwyl ek hierdie onverteenwoordigings verken, sal ek demonstreer dat die stel van on- verteenwoordigings van S nie bloot ’n versameling strukture is nie, maar self ’n algebraïese struktuur het. Dit is ’n hoop, en hierdie hoop van onverteenwoordigings is nou verwant aan die groep van ’interne simmetrieë’ van enige onverteenwoordiging van S. Hierdie insig stel ons in staat om die konsep van ’n omkeerbare element na die ryk van semigroepe uit te brei. Benewens hierdie grondliggende werk, ondersoek ek ook twee spesiale gevalle van onverteenwoordiging: In die konteks van Clifford semigroepe toon ek hoe die onverteenwoordiging van ’n struktuur in verskeie ’kleiner’ onverteenwoordigings van komponentstrukture opgebroke kan word (saam met ’n voorwaarde wat spesifiseer hoe hierdie kleiner onverteenwoordigings moet interaksie hê). Verder, in die konteks van kategorie-teorie, illustreer ek hoe die konsep van voorstelling (en dus onverteenwoordiging) veralgemeen kan word - nie van ’n groep na ’n semigroep nie, maar van ’n stel na ’n grafiek. Masters 2025-02-05T14:23:05Z 2025-02-05T14:23:05Z 2024-12 Thesis https://scholar.sun.ac.za/handle/10019.1/131689 en Stellenbosch University vii, 64 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Cayley algebras Monoids Buildings (Group theory) Semigroups UCTD Joubert, Gideo On reverse representation |
| title | On reverse representation |
| title_full | On reverse representation |
| title_fullStr | On reverse representation |
| title_full_unstemmed | On reverse representation |
| title_short | On reverse representation |
| title_sort | on reverse representation |
| topic | Cayley algebras Monoids Buildings (Group theory) Semigroups UCTD |
| url | https://scholar.sun.ac.za/handle/10019.1/131689 |
| work_keys_str_mv | AT joubertgideo onreverserepresentation |