[Haskell-cafe] Endo a, endomorphisms

[Adrian]

‐‐‐‐‐‐‐ Original Message ‐‐‐‐‐‐‐

On Sunday, December 5th, 2021 at 8:53 AM, Tony Zorman <tonyzorman at mailbox.org> wrote:

> On Sun, Dec 05 2021 14:39, Adrian via Haskell-Cafe wrote:
>
> > 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. As the examples I presented
above show, even stating that function composition is an endomorphism of Endo a (endofunctions)
seems incorrect.