Exercise 11, § 1

A lattice E is said to be complete if every subset of E has a least upper bound and a greatest lower bound in E; this means, in particular, that E has a greatest and a least element.

(a) Show that if an ordered set E is such that every subset of E has a least upper bound in E, then E is a complete lattice.

(b) A product of ordered sets is a complete lattice if and only if each of the factors is a complete lattice.

(c) An ordinal sum (Exercise 3) $\sum_{\iota \in I} E_\iota$ is a complete lattice if and only if the following conditions are satisfied :

(I) I is a complete lattice.

(II) If J is a subset of I which has no greatest element, and if $\sigma = \sup J$, then $E_\sigma$ has a least element.

(III) For each $\iota \in I$ every subset of $E_\iota$ which has an upper bound in $E_\iota$ has a least upper bound in $E_\iota$.

(IV) For each $\iota \in I$ such that $E_\iota$ has no greatest element, the set of all $\varkappa > \iota$ has a least element $\alpha$, and $E_\alpha$ has a least element.

(d) The ordered set $\mathscr{A}(E, F)$ of increasing mappings of an ordered set E into an ordered set F (Exercise 6) is a complete lattice if and only if F is a complete lattice.