2013 | OriginalPaper | Chapter
On WQO Property for Different Quasi Orderings of the Set of Permutations
Authors : Sandra Ose, Juris Viksna
Published in: Mathematical and Engineering Methods in Computer Science
Publisher: Springer Berlin Heidelberg
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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
$\preccurlyeq$
.