Full Text Available
Note: Clicking the button above will open the full text document at the original institutional repository in a new window.
Categorical semi-direct products in varieties of groups with multiple operators
The notion of a categorical semidirect product was introduced by Bourn and Janelidze as a generalization of the classical semidirect product in the category of groups. The main aim of this work is to study the general properties of semidirect products of groups with operators, describe them in vario...
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | English |
| Published: |
Department of Mathematics and Applied Mathematics
2014
|
| Subjects: | |
| Tags: |
No Tags, Be the first to tag this record!
|
| _version_ | 1867613857504034816 |
|---|---|
| access_status_str | Open Access |
| author | Inyangala, Edward Buhuru |
| author2 | Janelidze, George |
| author_browse | Inyangala, Edward Buhuru Janelidze, George |
| author_facet | Janelidze, George Inyangala, Edward Buhuru |
| author_sort | Inyangala, Edward Buhuru |
| collection | Thesis |
| description | The notion of a categorical semidirect product was introduced by Bourn and Janelidze as a generalization of the classical semidirect product in the category of groups. The main aim of this work is to study the general properties of semidirect products of groups with operators, describe them in various classical varieties of such algebraic structures and apply the results to homological algebra and related areas of modern algebra. The context in which the study is done is a semiabelian category (that is, a pointed, Barr-exact and Bourn-protomodular category). The main result in the thesis is the construction of the semidirect product in a variety -RLoop of right -loops as the product of underlying sets equipped with the -algebra structure. A variety of right -loops is a variety that is pointed, has a binary + (not necessarily associative or commutative) and a binary satisfying the identities 0 + x = x, x + 0 = x, (x + y) y = x and (x - y) + y = x. Thus, -RLoop is a generalization of the variety of -groups introduced by Higgins and the results obtained are valid for varieties of -loops. We also describe precrossed and crossed modules in the variety -RLoop. The theory of crossed modules developed is independent of that developed by Janelidze for crossed modules in an arbitrary semiabelian category and gives simplified explicit formulae for crossed modules in -RLoop. Finally, we mention that our constructions agree with the known ones in the familiar algebraic categories, specifically the categories of groups, rings and Lie algebras. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/4890 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:42:48.866Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2014 |
| publishDateRange | 2014 |
| publishDateSort | 2014 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/4890 Categorical semi-direct products in varieties of groups with multiple operators Inyangala, Edward Buhuru Janelidze, George Mathematics and Applied Mathematics The notion of a categorical semidirect product was introduced by Bourn and Janelidze as a generalization of the classical semidirect product in the category of groups. The main aim of this work is to study the general properties of semidirect products of groups with operators, describe them in various classical varieties of such algebraic structures and apply the results to homological algebra and related areas of modern algebra. The context in which the study is done is a semiabelian category (that is, a pointed, Barr-exact and Bourn-protomodular category). The main result in the thesis is the construction of the semidirect product in a variety -RLoop of right -loops as the product of underlying sets equipped with the -algebra structure. A variety of right -loops is a variety that is pointed, has a binary + (not necessarily associative or commutative) and a binary satisfying the identities 0 + x = x, x + 0 = x, (x + y) y = x and (x - y) + y = x. Thus, -RLoop is a generalization of the variety of -groups introduced by Higgins and the results obtained are valid for varieties of -loops. We also describe precrossed and crossed modules in the variety -RLoop. The theory of crossed modules developed is independent of that developed by Janelidze for crossed modules in an arbitrary semiabelian category and gives simplified explicit formulae for crossed modules in -RLoop. Finally, we mention that our constructions agree with the known ones in the familiar algebraic categories, specifically the categories of groups, rings and Lie algebras. 2014-07-31T08:07:12Z 2014-07-31T08:07:12Z 2010 Doctoral Thesis Doctoral PhD http://hdl.handle.net/11427/4890 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Mathematics and Applied Mathematics Inyangala, Edward Buhuru Categorical semi-direct products in varieties of groups with multiple operators |
| thesis_degree_str | Doctoral |
| title | Categorical semi-direct products in varieties of groups with multiple operators |
| title_full | Categorical semi-direct products in varieties of groups with multiple operators |
| title_fullStr | Categorical semi-direct products in varieties of groups with multiple operators |
| title_full_unstemmed | Categorical semi-direct products in varieties of groups with multiple operators |
| title_short | Categorical semi-direct products in varieties of groups with multiple operators |
| title_sort | categorical semi direct products in varieties of groups with multiple operators |
| topic | Mathematics and Applied Mathematics |
| url | http://hdl.handle.net/11427/4890 |
| work_keys_str_mv | AT inyangalaedwardbuhuru categoricalsemidirectproductsinvarietiesofgroupswithmultipleoperators |