On WQO Property for Different Quasi Orderings of the Set of Permutations

Abstract

The property of certain sets being well quasi ordered (WQO) has several useful applications in computer science – it can be used to prove the existence of efficient algorithms and also in certain cases to prove that a specific algorithm terminates.

One of such sets of interest is the set of permutations. The fact that the set of permutations is not WQO has been rediscovered several times and a number of different permutation antichains have been published. However these results apply to a specific ordering relation of permutations $\preccurlyeq$, which is not the only ’natural’ option and an alternative ordering relation of permutations $\trianglelefteq$ (more related to ’graph’ instead of ’sorting’ properties of permutations) is often of larger practical interest. It turns out that the known examples of antichains for the ordering $\preccurlyeq$ can’t be used directly to establish that $\trianglelefteq$ is not WQO.

In this paper we study this alternative ordering relation of permutations $\trianglelefteq$ and give an example of an antichain with respect to this ordering, thus showing that $\trianglelefteq$ is not WQO. In general antichains for $\trianglelefteq$ cannot be directly constructed from antichains for $\preccurlyeq$, however the opposite is the case – any antichain for $\trianglelefteq$ allows to construct an antichain for $\preccurlyeq$.