When $(a + b)σ$ = $aσ + bσ$, we say $σ$ distributes over $+$.
- Ordinary multiplication distributes over ordinary summation.
- In a vector field, application of a linear function distributes over vector summation.
It is very nice to have a word for this property.
Another time we may see a rule such that: $$(a \times b)τ = aτ \times b + a \times bτ + aτ \times bτ$$ − Or that: $$(a + b)τ = aτ \times bτ^- + aτ^- \times bτ + 2 \times a \times b$$
A derivative of a product of functions (before we apply the Transcendental law of homogeneity) obeys the first variant: $(a b)' = a' b + a b' + a' b'$.
Ordinary square of ordinary sum obeys the second variant: $(a^1 + b^1)^2 = a^2b^0 + a^0b^2 + 2a^1b^1$.
These two examples keep following a similar pattern if we apply $τ$ further, iteratively. Visually, they both have something to do with a slicing of a hyper-interval.
Is there a name for rules like these?