Well, Better and In-Between

Abstract

Starting from well-quasi-orders (wqos), we motivate step by step the introduction of the complicated notion of better-quasi-order (bqo). We then discuss the equivalence between the two main approaches to defining bqo and state several essential results of bqo theory. After recalling the rôle played by the ideals of a wqo in its bqoness, we give a new presentation of known examples of wqos which fail to be bqo. We also provide new forbidden pattern conditions ensuring that a quasi-order is a better quasi-order.