Exercise 3, § 1

Let I be an ordered set and let $(\mathrm{E}_\iota)_{\iota \in \mathrm{I}}$ be a family of non-empty ordered sets indexed by I.

(a) Let F be the sum (Chapter II, § 4, no. 8) of the family $(\mathrm{E}_\iota)_{\iota \in \mathrm{I}}$; for each $x \in \mathrm{F}$, let $\lambda(x)$ be the index $\iota$ such that $x \in \mathrm{E}_\iota$; and let G be the graph consisting of all pairs $(x, y) \in \mathrm{F} \times \mathrm{F}$ such that either $\lambda(x) < \lambda(y)$ or else $\lambda(x) = \lambda(y)$ and $x \leqslant y$ in $\mathrm{E}_{\lambda(x)}$. Show that G is the graph of an ordering on F. The set F endowed with this ordering is called the ordinal sum of the family $(\mathrm{E}_\iota)_{\iota \in \mathrm{I}}$ (relative to the ordering on I) and is denoted by $\sum_{\iota \in \mathrm{I}} \mathrm{E}_\iota$. Show that the equivalence relation corresponding to the partition $(\mathrm{E}_\iota)_{\iota \in \mathrm{I}}$ of F satisfies conditions (C) and (C′) of Exercise 2, and that the quotient ordered set (Exercise 2) is canonically isomorphic to I.

(b) If the set I is the ordinal sum of a family $(\mathrm{J}_\lambda)_{\lambda \in \mathrm{L}}$ of ordered sets, where L is an ordered set, show that the ordered set $\sum_{\iota \in \mathrm{I}} \mathrm{E}_\iota$ is canonically isomorphic to the ordinal sum $\sum_{\lambda \in \mathrm{L}} \mathrm{F}_\lambda$, where $\mathrm{F}_\lambda = \sum_{\iota \in \mathrm{I}_\lambda} \mathrm{E}_\iota$ ("associativity" of the ordinal sum). If I is the linearly ordered set $\{1, 2\}$, we write $\mathrm{E}_1 + \mathrm{E}_2$ for the ordinal sum of $\mathrm{E}_1$ and $\mathrm{E}_2$. Show that $\mathrm{E}_2 + \mathrm{E}_1$ and $\mathrm{E}_1 + \mathrm{E}_2$ are not necessarily isomorphic.

(c) An ordinal sum $\sum_{\iota \in \mathrm{I}} \mathrm{E}_\iota$ is right directed if and only if I is right directed and $\mathrm{E}_\omega$ is right directed for each maximal element $\omega$ of I.

(d) An ordinal sum $\sum_{\iota \in \mathrm{I}} \mathrm{E}_\iota$ is totally ordered if and only if I and each $\mathrm{E}_\iota$ is totally ordered.

(e) An ordinal sum $\sum_{\iota \in \mathrm{I}} \mathrm{E}_\iota$ is a lattice if and only if the following conditions are satisfied :

(I) The set I is a lattice, and for each pair $(\lambda, \mu)$ of non-comparable indices in I, $\mathrm{E}_{\sup (\lambda, \mu)}$ (resp. $\mathrm{E}_{\inf (\lambda, \mu)}$) has a least (resp. greatest) element.

(II) For each $\alpha \in \mathrm{I}$ and each pair $(x, y)$ of elements of $\mathrm{E}_\alpha$ such that the set $\{x, y\}$ is bounded above (resp. bounded below) in $\mathrm{E}_\alpha$, the set $\{x, y\}$ has a least upper bound (resp. greatest lower bound) in $\mathrm{E}_\alpha$.

(III) For each $\alpha \in \mathrm{I}$ such that $\mathrm{E}_\alpha$ contains a set of two elements which has no upper bound (resp. lower bound) in $\mathrm{E}_\alpha$, the set of indices $\lambda \in \mathrm{I}$ such that $\lambda > \alpha$ (resp. $\lambda < \alpha$) has a least element (resp. greatest element) $\beta$, and $\mathrm{E}_\beta$ has a least element (resp. greatest element).