 Barrier in better-quasi-order theory

A barrier on an infinite $A\subseteq \mathbb{N}$ is a family of nonempty finite increasing sequences, viewed as finite subsets, no one contained properly in another, such that every infinite subset of $A$ has a member of the family as an initial segment. Write $s◃t$ if there is a finite increasing sequence $u$ of which $s$ is an initial segment and of which $t$ is an initial segment after its first term is deleted. This shift relation underlies the good-array definition of better-quasi-ordering.