Topological protomodular algebras

Abstract

Topological groups have very striking properties, which have already been generalized to weaker "group like" structures, like various kinds of loops. This paper intends to show evidence that this generalization holds for a much wider class of theories, known as the protomodular theories, and which admit both an elegant categorical characterization and an easy description in universal algebra terms. Thus we propose a synthetic approach which allows to prove in a unique framework that the most striking properties of topological groups hold as well for loops or even semi-loops, rings with or without unit, associative algebras with or without unit, Lie algebras, Jordan algebras, Boolean algebras, Heyting algebras, Boolean rings, Heyting semi-lattices, and so on.