Re: categories with several compositions?

["Fred E.J. Linton" <fejlinton@usa.net>]

On Wed, 02 Feb 2011 09:20:11 AM EST, John Stell <J.G.Stell@leeds.ac.uk>
asked:

> 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 a single identity that works for both and where
> (x *i y) *j z = x *i (y *j z) for all i,j ...

What am I missing when I think I see, using y = 1 (an identity map,
as suggested by the additional remarks Stell makes below), that

x *j z = (x *i 1) *j z = x *i (1 *j z) = x *i z ?

Cheers, -- Fred
-- 
> 
> More generally, a kind of category with several compositions:
> for each object y there is a set Dy and instead of the usual
> 
> C(x,y) x C(y,z) -> C(x,z)
> 
> we have   Dy -> [C(x,y) x C(y,z), C(x,z)]
> 
> So you have a family of compositions at each object which "associate with
> each other" in the manner of the above equation, and where there is
> a single identity for each object.
> 
> thanks
> John Stell



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