I'm trying to find a combinator that can derive all other combinators, so there is an expression in terms of this combinator (I'll call it U, for universal), for any (computable) combinator. I also want a proper combinator, which means that Uabc..., will be entirely in terms of a,b,c, etc. With such a combinator, I can make a new minimalist version of the Xi function. I've worked out a few properties such a combinator must posess:
1) The Cancellation Effect, which means when Uabc... is betareduced, one of the variables will disappear. As an example, the combinator K is defined as: Kxy=x, so y disaapears. No combinator with the cancellative property can be derived solely from combinators without said property.
2) The Duplicative Effect, which essentially means it duplicates one of the variables that are applied to it. As an example, the combinator M is defined as Mx=xx, so it has a duplicative effect on x.
3) The Composing Effect, which means that it composes some of the variables. As an example, the combinator B is defined as Bxyz=x(yz), so it composes x with y.
4) The Swapping Effect: When you beta reduce Uabc..., some of the variables are in the wrong order. For example, the combinator T is defined as: Txy=yx, so the x and y have been swapped around.
I have proved that any complete combinator has arity at least 3, and that Uabc is not a function of just a and b. Right now I'm working on the combinator A defined by Aabc=a(ca).