Combinatory algebrasIn algebra the nature of elements (henceforth "objects") is irrelevant. Nevertheless, it helps to have some notion of "object" in mind to get a taste of combinatory algebra. One such notion considers an "object" as being any set of properties expressed in some (first order) language. If that lacks imagery, then simply consider as "object" anything that can be object of thought (cf. Engeler 1995). What can be said about a universe C based on such an abstract notion of "object" and some (as yet entirely unspecified) notion of "multiplication" within C? A number of things, as it turns out.
To make an example, take three particular objects in C, say a, b, c, and combine them in a certain way, say, (a·(b·a))·c. By following the "multiplication" table of C this combination will result in, say, z. Now, any three objects could be combined in this way, and "to combine arbitrary three objects, x1, x2, x3 into (x1·(x2·x1))·x3" is itself a (first order) property. As such it specifies some object that must, by definition, exist in C. Call that object P. The defining property of P is precisely that for its "multiplication" with any three objects x1, x2, x3 the relationship ((((P·x1)·x2)·x3) = (x1·(x2·x1))·x3 holds in C. By this reasoning we conclude that for any arbitrary combination of an arbitrary number of objects in C there must be another object that expresses the possibility of combining things in that way, and there will be a corresponding relationship (an equality) stating that fact. This remarkable property is known as "combinatory abstraction". A series of other properties follow from it, but we won't pursue this matter further. Universes in which it holds are "combinatory complete", and are known as "combinatory algebras".