categories with several compositions?

[John Stell <J.G.Stell@leeds.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 a single identity that works for both and where
(x *i y) *j z = x *i (y *j z) for all i,j

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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