Ordinal sum of posets

Remarks

If $P$ and $Q$ are not disjoint, first replace them by disjoint copies. Example: with $P = \set{a}$ and $Q = \set{b, c}$ an antichain, $P \oplus Q$ has $a < b$ and $a < c$ by clause 3, and $b$, $c$ stay incomparable. Non-example: in $Q \oplus P$ the element $a$ is above $b$ and $c$, so $Q \oplus P$ differs from $P \oplus Q$; the ordinal sum is not commutative. The ordinal sum is associative, and the ordinal sum of two total orders is a total order. For well-orders, it gives the sum of ordinal numbers, so $\omega \oplus \set{x}$ has order type $\omega + 1$.