[Haskell-cafe] Endo a, endomorphisms

[Tom Ellis]

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

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