# From Well-Quasi-Ordered Sets to Better-Quasi-Ordered Sets

### Abstract

We consider conditions which force a well-quasi-ordered poset (wqo) to be better-quasi-ordered (bqo). In particular we obtain that if a poset $P$ is wqo and the set $S_{\omega}(P)$ of strictly increasing sequences of elements of $P$ is bqo under domination, then $P$ is bqo. As a consequence, we get the same conclusion if $S_{\omega} (P)$ is replaced by ${\cal J}^{\neg \downarrow\hskip -2pt }(P)$, the collection of non-principal ideals of $P$, or by $AM(P)$, the collection of maximal antichains of $P$ ordered by domination. It then follows that an interval order which is wqo is in fact bqo.