Semiorder

Remarks

Clause 1 forbids two incomparable two-element chains, and clause 2 forbids a three-element chain with an element incomparable to all three of its elements. Example: on $\R$, let $x \prec y$ when $x + 1 < y$. For clause 2, if $a + 1 < b$ and $b + 1 < c$ but $d \le a + 1$, then $d + 1 \le a + 2 < c$, so $d \prec c$; clause 1 has a similar check. Non-example: the poset with $a \prec b \prec c$ and an element $d$ incomparable to $a$, $b$ and $c$ satisfies clause 1 but fails clause 2. Every semiorder is an interval order. By a theorem of Scott and Suppes, a finite strict partial order is a semiorder if and only if there is a map $u \colon X \to \R$ with $x \prec y \iff u(x) + 1 < u(y)$.