RE: categories with several compositions?

[N.Bowler@dpmms.cam.ac.uk]

> Can anyone tell me whether these structures have been studied anywhere?
>
> A kind of generalized monoid with two or more compositions *1, *2, etc
> with [each composition having its own identity] and where
> (x *i y) *j z = x *i (y *j z) for all i,j
We can certainly relate these structures to things we already understand
reasonably well (though I can't see how to say `they are just wombats').

First of all, here's a way to generate lots of examples. Pick any monoid M,
and let (e_i | i is in I) be any family of invertible elements of M. For
each i in I, define a new operation *i on M by a *i b = a * (e_i^{-1}) * b.
M is a monoid under each *i, with identities the s_i, and (x *i y) *j z = x
*i (y *j z) for all i,j.

In fact, all examples arise in this way. Pick one of the operations, which
we will treat differently from the others: call it * and call its identity
e. So the structure is a monoid with respect to * and e. I'll call this
monoid M. Now pick some other operation *i and let e_i be the identity for
*i.

e_i * (e *i e) = (e_i * e) *i e = e_i *i e = e

and similarly (e *i e) * e_i = e, so e_i is an invertible element of M with
inverse (e *i e). Now note that for any a and b,

a *i b = (a * e) *i (e * b) = a * (e *i e) * b = a * (e_i^{-1}) * b

so that each *i arises as above.

On the basis of this analysis, it looks like the structures you asked about
bear the same sort of relation to monoids with a designated family of
invertible elements that torsors do to groups.

Nathan


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