Re: [Haskell-cafe] Endo a, endomorphisms

On Sun, Dec 05, 2021 at 03:24:13PM +0000, Adrian via Haskell-Cafe wrote:

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On Sunday, December 5th, 2021 at 8:53 AM, Tony Zorman wrote:

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On Sun, Dec 05 2021 14:39, Adrian via Haskell-Cafe wrote:

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According to Algebra [Hungerford 74], an endomorphism is an

endofunction that is a homomorphism. A set of endomorphisms is quite

distinct from a set of endofunctions in this regard.

What counts as a "homomorphism" is very dependent on the context that

you're in. Here, we are not studying some exotic algebraic structure,

but really just functions over a set. In particular, "(homo)morphism"

becomes an alias for "function".

I note that in the paper "Monoid: Theme and Variations" [Yorgey 2012], a monoid homomorphism is defined in a manner consistent with the definitions found in Algebra [Hungerford 74]:

A monoid homomorphism is a function from one monoidal type to another which preserves monoid structure; that is, a function f satisfying the laws:

f ε = ε f (x <> y) = f x <> f y

So again, given that the context is the Data.Monoid library, it seems much more appropriate to say that Endo a forms a monoid of endofunctions under composition.

Yes, I agree that would be a clarifying rewrite, avoiding a clash of terminology.