Exercise 14, § 1

Algebra, VI, p. 32

Show that, in a semi-lattice ordered monoid $M$ .

\inf(x_i + y_i) \geq \inf(x_i) + \inf(y_i)

for all finite sequences of n terms $(x_i), (y_i)$ . Deduce the inequalities

(x + y)^+ \leq x^+ + y^+ , \quad |x^+ - y^+| \leq |x - y|

in any lattice ordered group.

Solution

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