Remark

Tag 06XO, permanent. Cite it as [Bourbaki, Tag 06XO].

Algebra, chapter VI, § 1. Ordered groups. Divisibility

Applying the formulae of Prop. 13 to a family of two elements $(x, y)$ , we obtain the following formulae, which express the fact that each of the laws of composition sup, inf in a lattice ordered group is distributive with respect to the other :

\begin{align*} \sup(z, \inf(x, y)) &= \inf(\sup(z, x), \sup(z, y)) \\ \inf(z, \sup(x, y)) &= \sup(\inf(z, x), \inf(z, y)) . \end{align*}

This distributivity property is peculiar to lattice ordered groups, and does not extend to arbitrary lattices, or even to lattice ordered monoids (cf. VI, p. 33, Ex. 24).

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

Cited by

Nothing.

Cites

Prop. 13