On why addition of ultrafilters over N is not commutative

[Bertalan Pecsi <zellerede@gmail.com>]

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Hi,

Towards an understanding of the phenomenon why taking ultrafilters carry over binary operations and associativity but not commutativity (in the sense that if S is a semigroup then it induces a semigroup structure on βS), first I succeeded to generalize this to all monads over Set  - I guess it should be well known as well, but then realized that actually the unit of the monad is not required in the proof, so actually all 'semigroupads' (or 'semimonads') carries over semigroups.
The unit of the monad is required exactly to carry over monoids.

I played around with this idea, and could also come up with correspondent identities involving the Kleisli star in place of the binary monad operation.
In particular, a magmad on a category  C  is just an endofunctor  T:C->C  and a nat.transformation  μ:TT->T  (without any further requirements).
For example, a 'central element' of (T,μ) would be a nat.transf. ζ:1->T satisfying μ ζT = μ Tζ, and in this case any magma S with central element c, we'll have ζ(c) central in the induced magma TS.

So, I basically found some answer to the question in the title: the ultrafilter monad itself is not commutative (whatever it should mean(?)) and therefore commutativity is not carried over.

Does anyone know about these kinds of ideas appearing in the literature?

Thanks in advance,
   Bertalan Pécsi


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