Well Quasi Ordering Finite Posets and Formal Languages

We show that the set of finite posets is a well-quasi-ordering with respect to a certain relation ⪯, called the chain minor relation. To prove this we introduce a similar relation on finite formal languages which also has this property. As a consequence we get that every property which is hereditary with respect to ⪯ has a test in O(|P|c), whereas c depends on the property. This test has an easy parallelization with the same costs. On a parallel machine (CRCW PRAM) it may be implemented in such a way that it runs in constant time and needs O(|P|c)processors.