WQO - why transitivity?

Question

This article says that $\sf WQO$s are a strenghtening of the definition of well-founded relations for the purpose of having the property be preserved by operations. However, the few operations and proofs (minimal bad sequence arguments) that I know about don't seem to need the extra assumption of transitivity. The only exception is the product of relations/posets but it can be done with infinite Ramsey's theorem. Is transitivity used in other parts of $\sf WQO/BQO$ theory?

Answer 1

This likely isn't the greatest theorem of WQO theory but for an example, a qoset is WQO iff it has no infinite decreasing chains nor infinite antichains. This fails for WQO-minus-transitivity, consider the relation $\preceq$ on $\mathbb{N}$ such that $a \preceq b \iff a=b \lor a=b+1 \lor (a<b \land a\equiv b \mod 2)$. There is an infinite decreasing chain $0\succ1\succ2\ldots$ but in an infinite sequence we will eventually have two numbers of same parity in increasing order.