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As I understand it, given a set $A$, an endomorphism is a function $f$ which maps $A$ to itself. $f : A \rightarrow A$

So, for a concrete example, would we consider a permutation matrix an endomorphism? or, maybe just the function $\forall x \in A, f(x) = x$?

Blake
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    It depends what category you are in, but an endomorphism is defined as a structure preserving map from a structure to itself. For instance, in the category of sets, there is no structure to preserve so the endomorphisms are simply maps from sets to themselves. – Edward Evans Apr 14 '17 at 20:46
  • Possible duplicate, possibly useful: https://math.stackexchange.com/questions/2039702/what-is-an-homomorphism-isomorphism-saying/2039715#2039715 – Ethan Bolker Apr 14 '17 at 20:55
  • Oh, I think I see now. So, if I had a group like $\mathbb{R}^3 $ with $ [u, v] = u\times v$, an endomorphism would preserve that structure or just the lie algebra? – Blake Apr 14 '17 at 21:00
  • Cross product isn't a group operation, it's not associative – D_S Apr 14 '17 at 21:01
  • A group endomorphism just means a group homomorphism from a group to itself. Similarly ring endomorphism, etc. – D_S Apr 14 '17 at 21:01

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As implied in one of the comments, context matters.

A permutation matrix is an endomorphism when acting on the set of all orderings of a finite collection of items. But for example, the permutation $S_{213}$ that swaps the first and second element of an ordered $n$-tuple is not an endomorphisim acting on the set $\{(1,2,3), (2,3,1), (3,1,2)\}$.

Your second example, where you have specified the set $A$, is indeed always an endomorphism. It works on every element of the set $A$, and everything it can transform an element into is indeed an element of $A$.

Mark Fischler
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