Bourbaki

Exercise 10, § 1

Algebra, VI, p. 31

Show that, for every finite sequence (xi)(x_i) (1 ≤ i ≤ n) of elements of a lattice ordered group G,

sup⁡(xi)=∑ixi−∑i<jinf⁡(xi,xj)+⋯+(−1)p+1∑i1<i2<...<ipinf⁡(xi1,...,xip)+⋯+(−1)n+1inf⁡(x1,...,xn). \sup(x_i) = \sum_i x_i - \sum_{i < j} \inf(x_i, x_j) + \cdots + (-1)^{p+1} \sum_{i_1 < i_2 < ... < i_p} \inf(x_{i_1}, ..., x_{i_p}) + \cdots + (-1)^{n+1}\inf(x_1, ..., x_n).

(Argue by induction based on Prop. 7 (VI, p. 10), using the distributivity of sup over inf.)

Solution

No solution has been written yet.

Transcribed from the printed volume. Not translated.