The purpose of this seminar is to introduce the notion of ideals of well-quasi-orders. These irreducible downwards-closed sets of elements were first invented in the 1970’s but rediscovered in recent years in the theory of well-structured transition systems, notably by Finkel and Goubault-Larrecq. Ideals provide indeed finite effective representations of downwards-closed sets, in the same way as bases of minimal elements provide representations of upwards-closed sets.

After defining ideals and establishing some of their properties, I will illustrate their use in a concrete setting. I will present some recent results by Czerwiński, Martens, van Rooijen, and Zeitoun (FCT’15) on the decidability of piecewise-testable separability in the light of ideals. This seminar is also a warm-up for the next seminar on reachability in vector addition systems.