On WQO Property for Different Quasi Orderings of the Set of Permutations
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 \(\preccurlyeq\).
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