Definition 7

Tag 0585, permanent. Cite it as [Bourbaki, Tag 0585].

Algebra, chapter I, § 3. Actions

Let $T$ and $\perp$ be two internal laws on a set E. The law $\perp$ is said to be distributive with respect to the law $T$ if

(10)

x \perp (y T z) = (x \perp y) T (x \perp z)

(11)

(x T y) \perp z = (x \perp z) T (y \perp z)

for all

x, y, z

in E.

Note that (10) and (11) are equivalent if the law $\perp$ is commutative. In general, one of the laws is written additively and the other multiplicatively; if multiplication is distributive with respect to addition, then:

(12)

x . (y + z) = x . y + x . z

(13)

(x + y) . z = x . z + y . z

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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