Distributive Products and Their Homology

We develop a theory of sets with distributive products (called shelves and multi-shelves) and of their homology. We relate the shelf homology to the rack and quandle homology. 1. Binary distributive operations Let X be a set. Binary operations ? : X ×X → X with composition x ?1 ?2y = (x ?1 y) ?2 y and the identity x ? y = x form a monoid, M(X). A collection of operations ?i, i ∈ I is called mutually (right) distributive if (a ?i b) ?j c = (a ?j c) ?i (b ?j c) for all i, j ∈ I. A direct computation gives: Lemma 1. If {?i}i∈I are mutually-distributive then the set of all compositions of ?i’s is mutually distributive. Hence every maximal set of mutually-distributive products is a monoid. We say that ? is (right) self-distributive or, simply, distributive, if (1) (x ? y) ? z = (x ? z) ? (y ? z), that is, if the one element set {?} is mutually distributive. In this case, (X, ?) is called a shelf, [Cr]. (Hence (1) might be called “shelf-distributivity”.) We abbreviate x ? y to xy whenever it does not lead to a confusion. We call a collection of mutually distributive products on a set, (X, ?i, i ∈ I), a multi-shelf. Example 2. Every set X with the one of the following products is a shelf: (1) x ? y = f(x), for some f : X → X. (2) x ? y = g(y) for some g : X → X such that g = g. (3) Given A ⊂ X, x ? y = { y if y ∈ A x otherwise. Example 3. Let X be a collection of subsets of Ω. (1) If X is closed under intersections then (X,∩) is a shelf. (2) If X is closed under set subtractions then X with operation of subtraction is a shelf. (3) X = 2 with operations ∩,∪, and the “identity” product, x ? y = y, is a multishelf.