Exercise 16, § 1

A lattice E is said to be distributive if it satisfies the following two conditions :

(D') $\qquad \sup (x, \inf (y, z)) = \inf (\sup (x, y), \sup (x, z)),$ (D'') $\qquad \inf (x, \sup (y, z)) = \sup (\inf (x, y) \inf (x, z))$

for all $x, y, z$ in E. A totally ordered set is a distributive lattice.

(a) Show that each of the conditions (D), (D') separately implies the condition

(D) $\quad \sup (\inf (x, y), \ \inf (y, z), \ \inf (z, x))$

for all $x, y, z$ in E.

(b) Show that the condition (D) implies the condition

(M) $\qquad$ If $x \geqslant z$, then $\sup (z, \inf (x, y)) = \inf (x, \sup (y, z)).$

Deduce that (D) implies each of (D′) and (D″), and hence that the three axioms (D), (D′), and (D″) are equivalent (to show, for example, that (D) implies (D′), take the least upper bound of $x$ and each side of (D), and use (M)).

(c) Show that each of the two conditions

(T′) $\qquad\qquad \inf\,(z, \sup\,(x, y)) \leqslant \sup\,(x, \inf\,(y, z)),$ (T″) $\inf\,(\sup\,(x, y), \sup\,(z, \inf\,(x, y))) = \sup\,(\inf\,(x, y), \inf\,(y, z), \inf\,(z, x))$

for all $x$, $y$, $z$ in E is necessary and sufficient for E to be distributive. (To show that (T′) implies (D″), consider the element