 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.