Interval order

Remarks

Example: the poset on $\set{a, b, c}$ with $a < b$ as the only strict relation is an interval order. The intervals $[0, 1]$, $[2, 3]$ and $[0.5, 2.5]$ for $a$, $b$, $c$ work: $1 < 2$ gives $a < b$, and the interval of $c$ meets both others. Non-example: the poset $\mathbf{2} + \mathbf{2}$ with $a < b$, $c < d$ and no other strict relation is not an interval order. Intervals would give $r_a < l_b$ and $r_c < l_d$, and also $l_d \le r_a$ and $l_b \le r_c$, so $r_a < l_b \le r_c < l_d \le r_a$, which is impossible. Fishburn's theorem says that a finite poset is an interval order if and only if it has no subposet isomorphic to $\mathbf{2} + \mathbf{2}$. Every total order is an interval order: on a finite total order, use the points $[i, i]$.