categories with several compositions?

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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]