Exercise 6, § 1

Let E and F be two ordered sets, and let $\mathscr{A}(\mathrm{E}, \mathrm{F})$ be the subset of the product ordered set $\mathrm{F}^{\mathrm{E}}$ consisting of the increasing mappings of E into F.

(a) Show that if E, F, G are there ordered sets, then the ordered set $\mathscr{A}(\mathrm{E}, \mathrm{F} \times \mathrm{G})$ is isomorphic to the product ordered set $\mathscr{A}(\mathrm{E}, \mathrm{F}) \times \mathscr{A}(\mathrm{E}, \mathrm{G})$.

(b) Show that if E, F, G are three ordered sets, then the ordered set $\mathscr{A}(\mathrm{E} \times \mathrm{F}, \mathrm{G})$ is isomorphic to the ordered set $\mathscr{A}(\mathrm{E}, \mathscr{A}(\mathrm{F}, \mathrm{G}))$.

(c) If $\mathrm{E} \neq \emptyset$, then $\mathscr{A}(\mathrm{E}, \mathrm{F})$ is a lattice if and only if F is a lattice.

(d) Suppose that E and F are both non-empty. Then $\mathscr{A}(\mathrm{E}, \mathrm{F})$ is totally ordered if and only if one of the following conditions is satisfied :

($\alpha$) F consists of a single element;

($\beta$) E consists of a single element and F is totally ordered;

($\gamma$) E and F are both totally ordered and F has two elements.