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A BQO, or better-quasi-ordering, strengthens a well-quasi-ordering by testing arrays indexed by arbitrary barriers, rather than just sequences. Let $Q$ be a preorder. A barrier in better-quasi-order theory on an infinite $A\subseteq \omega$ is a family $B$ of nonempty finite increasing sequences such that no member is properly contained in another and every infinite subset of $A$ has a member of $B$ as its initial segment.

For barrier members write $s◃t$ if a finite increasing sequence $u$ has $s$ as an initial segment and has $t$ as an initial segment after deleting the first entry of $u$. A map $f:B\to Q$ is good when some $s◃t$ satisfy $f(s)\le f(t)$. The definition isBoth quantifiers matter: testing only barriers of one fixed finite length is insufficient.

For the singleton barrier $B=[A{]}^1$, shifted pairs are just $\{a\}◃\{b\}$ with $a<b$. Thus the BQO condition implies the ordinary well-quasi-ordering condition. Barrier arrays can also be viewed through continuous arrays in better-quasi-order theory $F:[A{]}^{\omega }\to Q$, with $Q$ discrete: set $F(X)=f(s)$ for the unique barrier initial segment $s$ of $X$. This is continuous because that value is fixed by a finite prefix, and comparing $X$ with $X\setminus \{\min X\}$ is exactly the shift comparison. The general continuous-array formulation is equivalent to the barrier definition; its converse uses refinement of fronts to barriers, rather than the incorrect assertion that every prefix-decision family is already a barrier.

For an illustrative positive example, every well-order is a BQO. If a barrier array into a well-order were bad, its induced $F$ would satisfyfor every infinite $X$. This is an infinite strict descent in a well-order, a contradiction. A finite antichain with equality is also a BQO: the finitely many fibres of the induced continuous array are open, so repeated use of the Open Ramsey theorem gives an infinite $A^{\prime }$ on which the array is constant. The comparison of an infinite subset of $A^{\prime }$ with its shift is then good.

The implication to a well-quasi-order is strict. The Rado order isIt is well-quasi-ordered: if one first coordinate occurs infinitely often, its second coordinates have an increasing pair; otherwise the first coordinates are unbounded, and a later one exceeds the second coordinate of a chosen earlier pair. But take the barrier $[\omega {]}^2$ and $f(\{m,n\})=(m,n)$. For $m<n<l$, the shift compares $(m,n)$ with $(n,l)$; their first coordinates differ and $n<n$ is false. Every shifted comparison therefore fails. This bad barrier array for the Rado order proves well-quasi-ordering does not imply better-quasi-ordering.

The stronger notion is designed for infinitary closure operations, such as powerset and infinite-sequence constructions, where a mere well-quasi-order can develop new bad arrays. The barrier definition records variable finite amounts of information from an infinite object and is consequently more robust than checking sequences alone.