Category-theoretic Remarks
In category theory, a group with operators can be defined as an object of a functor category GrpM where M is a monoid (i.e., a category with one object) and Grp denotes the category of groups. This definition is equivalent to the previous one, provided is a monoid (otherwise we may expand it to include the identity and all compositions).
A morphism in this category is a natural transformation between two functors (i.e. two groups with operators sharing same operator domain M). Again we recover the definition above of a homomorphism of groups with operators (with f the component of the natural transformation).
A group with operators is also a mapping
where is the set of group endomorphisms of G.
Read more about this topic: Group With Operators
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