A compatibility law between two binary operations · and + on a set stating that a·(b + c) = a·b + a·c and (b + c)·a = b·a + c·a for all elements, so one operation distributes over the other.

Principle

Compatibility of composition and combination: one operation acts linearly over the structure formed by the other, allowing expansion and factorization of mixed expressions.

Demonstration

In any ring, multiplication distributes over addition: 2·(3+4) = 2·7 = 14 and 2·3 + 2·4 = 6 + 8 = 14; conversely, composition of linear maps distributes over addition of maps.

Misapplication

Assuming distributivity between arbitrary operations (for example that function composition distributes over multiplication of functions) without verification leads to incorrect expansions.

Consequence

Distributivity enables algebraic expansion, factoring, and the definition of ring-like structures; it is essential for polynomial arithmetic and for building linear structures over a base operation.

Reversal

Non-distributivity: when no consistent expansion law exists between the two operations; some algebraic systems are deliberately non-distributive to model different interactions (e.g., certain lattices or nonassociative algebras).

Boundary

Requires two specified operations and must be checked on the whole set; distributivity can be one-sided (left or right only) and may fail on certain subsets or for special elements.

Semantic Tension

Distributivity is distinct from linearity and homomorphism: linear maps satisfy a version of distributivity over scalar addition, but distributivity as a binary law concerns internal operations rather than external scalar action.

Synthesis

The distributive law ties two operations so that the action of one spreads over combinations formed by the other, making expansion and factorization coherent and enabling layered algebraic structures such as rings and modules.