I chose to begin with the notes out of which I constructed the central definition below. The equation which defines `distributivity' is:

a(b+c) = ab + ac

This has, of course, a `reversed' form, (b+c)a = ba+ca: I chose to name the displayed form `left' distributive and this latter form `right' distributive. When cast in the general terms of binary operators, naming multiplication f and addition g, we have, for any legitimate a, b and c:

f(a, g(b,c)) = g(f(a,b), f(a,c))

Thus, if we take (A×B|f:C) and (D×E|g:F) as temporary namings for the domains and ranges of our binary operators, we obtain

a is in A;
g(b,c) (in F), b and c are in B;
f(a,b) (in C) and b are in D; and
f(a,c) (in C) and c are in E.

so we need F to be a subset of B and C to be a subset of D and of E. I chose to take C=D=E=F=B for this left-distributive case, replacing B with A for right-distributive.

Distributivity

A binary operator, (A×B|f:B), left-distributes over a uniform binary operator, g, on B precisely if, for every a in A and b, c in B: f(a,g(b,c)) = g(f(a,b),f(a,c)). We say (B×A|f:B) right-distributes over (B×B|g:B) precisely if, for every a in A and b, c in B: f(g(b,c),a) = g(f(b,a),f(c,a)). One binary operator is said to distribute over another precisely if the former both left-distributes and right-distributes over the latter - in which case both are necessarily uniform and the two are parallel (that is, they act on the same space).

In particular, any Abelian binary operator which left- or right-distributes over some binary operator inevitably distributes over the latter. When B and A are distinct, (A×B|f:B) can only distribute from the left over anything, and that must be over some (B×B|:B), so there is no ambiguity in refering to such an f as distributing over some g, implicitly uniform on B. It should also be noted that if f does left-distribute over some g, then its transpose, (B×A| (b,a)->f(a,b) :B), right-distributes over g.

Distributivity

Further reading