Objects with an Algebraic Structure

Abstract

Algebraic structures on sets are created by “algebraic combinations” such as multiplication or addition of two elements, or formation of inverses and neutral elements with to respect to such a combination. We restrict ourselves here to operations that are defined everywhere rather than only for the elements of suitable subsets. This means in particular, that we leave out fields and division algebras, since for them inverses with respect to multiplication are not defined for ali elements. With this restriction, algebraic combinations can be described by maps from products, and the usual laws satisfied by such combinations can be described by commutative diagrams. This, however, is possible in an arbitrary category, provided the necessary products and a terminal object exist, which will always be assumed in the following. Dualization produces co-algebraic structures, where coproducts and initial objects are required. This does not yield any interesting examples in Ens, but it does in other categories.