A poset $\PXP$ is called an interval order if there exists a function $I$ assigning to each element $x\in X$ a closed interval $I(x)=[a_x,b_x]$ of the real line $\reals$ so that for all $x\text{,}$ $y\in X\text{,}$ $x\lt y$ in $P$ if and only if $b_x\lt a_y$ in $\reals\text{.}$ We call $I$ an interval representation of $\bfP\text{,}$ or just a representation for short. For brevity, whenever we say that $I$ is a representation of an interval order $\PXP\text{,}$ we will use the alternate notation $[a_x,b_x]$ for the closed interval $I(x)\text{.}$ Also, we let $|I(x)|$ denote the length of the interval, i.e., $|I(x)|=b_x-a_x\text{.}$ Returning to the poset $\bfP_3\text{,}$ the representation shown in Figure 6.28 shows that it is an interval order.
Note that end points of intervals used in a representation need not be distinct. In fact, distinct points $x$ and $y$ from $X$ may satisfy $I(x)=I(y)\text{.}$ We even allow degenerate intervals, i.e., those of the form $[a,a]\text{.}$ On the other hand, a representation is said to be distinguishing if all intervals are non-degenerate and all end points are distinct. It is relatively easy to see that every interval order has a distinguishing representation.
As we shall soon see, interval orders can be characterized succinctly in terms of forbidden subposets. Before stating this characterization, we need to introduce a bit more notation. By $\bfn$ (for $n\geq 1$ an integer), we mean the chain with $n$ points. More precisely, we take the ground set to be $\{0,1,\dots,n-1\}$ with $i \lt j$ in $\bfn$ if and only if $i\lt j$ in $\ints\text{.}$ If $\PXP$ and $\QYQ$ are posets with $X$ and $Y$ disjoint, then $\bfP+\bfQ$ is the poset $\bfR=(X\cup Y,R)$ where the partial order is given by $z\leq w$ in $R$ if and only if (a) $z,w\in X$ and $z\leq w$ in $P$ or (b) $z,w\in Y$ and $z\leq w$ in $Q\text{.}$ Thus, $\bfn+\bfm$ consists of a chain with $n$ points and a chain with $m$ points and no comparabilities between them. In particular, $\bftwo+\bftwo$ can be viewed as a four-point poset with ground set $\{a,b,c,d\}$ and $a\lt b$ and $c\lt d$ as the only relations (other than those required to make the relation reflexive).
We show only that an interval order cannot contain a subposet isomorphic to $\bftwo+\bftwo\text{,}$ deferring the proof in the other direction to the next section. Now suppose that $\PXP$ is a poset, $\{x,y,z,w\}\subseteq X$ and the subposet determined by these four points is isomorphic to $\bftwo+\bftwo\text{.}$ We show that $\bfP$ is not an interval order. Suppose to the contrary that $I$ is an interval representation of $\bfP\text{.}$ Without loss of generality, we may assume that $x\lt y$ and $z\lt w$ in $P\text{.}$ Thus $x\Vert w$ and $z\Vert y$ in $P\text{.}$ Then $b_x\lt a_y$ and $b_z \lt a_w$ in $\reals$ so that $a_w \le b_x \lt a_y \le b_z\text{,}$ which is a contradiction.