Categorical semi-direct products in varieties of groups with multiple operators - Inyangala, Edward Buhuru (Department of Mathematics and Applied Mathematics, 2014)
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.