Proposition 12

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

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

— Let $x_i$ ( $1 \leq i \leq n$ ) be n pairwise coprime elements in a lattice ordered group. Then

\sup(x_1, \ldots, x_n) = x_1 + \cdots + x_n .

This follows from the formula $u + v = \sup(u, v) + \inf(u, v)$ (VI, p. 10, Prop. 7) by induction on $n$ , using the fact that $x_i$ is coprime to $x_1 + \ldots + x_{i-1}$ for $2 \leq i \leq n$ (Cor. 4 to Prop. 11).

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

Cited by

Nothing.

Cites

Cor. 4 to Prop. 11