Categorical semi-direct products in varieties of groups with multiple operators

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dc.contributor.advisorJanelidze, Georgeen_ZA
dc.contributor.authorInyangala, Edward Buhuruen_ZA
dc.date.accessioned2014-07-31T08:07:12Z
dc.date.available2014-07-31T08:07:12Z
dc.date.issued2010en_ZA
dc.description.abstractThe 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.en_ZA
dc.identifier.apacitationInyangala, E. B. (2010). <i>Categorical semi-direct products in varieties of groups with multiple operators</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/4890en_ZA
dc.identifier.chicagocitationInyangala, Edward Buhuru. <i>"Categorical semi-direct products in varieties of groups with multiple operators."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2010. http://hdl.handle.net/11427/4890en_ZA
dc.identifier.citationInyangala, E. 2010. Categorical semi-direct products in varieties of groups with multiple operators. University of Cape Town.en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Inyangala, Edward Buhuru AB - 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. DA - 2010 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 2010 T1 - Categorical semi-direct products in varieties of groups with multiple operators TI - Categorical semi-direct products in varieties of groups with multiple operators UR - http://hdl.handle.net/11427/4890 ER - en_ZA
dc.identifier.urihttp://hdl.handle.net/11427/4890
dc.identifier.vancouvercitationInyangala EB. Categorical semi-direct products in varieties of groups with multiple operators. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2010 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/4890en_ZA
dc.language.isoeng
dc.publisher.departmentDepartment of Mathematics and Applied Mathematicsen_ZA
dc.publisher.facultyFaculty of Scienceen_ZA
dc.publisher.institutionUniversity of Cape Town
dc.subject.otherMathematics and Applied Mathematicsen_ZA
dc.titleCategorical semi-direct products in varieties of groups with multiple operatorsen_ZA
dc.typeDoctoral Thesis
dc.type.qualificationlevelDoctoral
dc.type.qualificationnamePhDen_ZA
uct.type.filetypeText
uct.type.filetypeImage
uct.type.publicationResearchen_ZA
uct.type.resourceThesisen_ZA
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