Abstract
A quasi-ordered setA (i.e. one equipped with a reflexive and transitive relation ≤) is said to be well-quasi-ordered (wqo) if for every infinite sequencea 1,a 2, ... of elements ofA there are indicesi, j such thati < j anda i ≤a j.
Various natural wqo setsQ admit “labelling” by another wqoA yielding another quasi-ordered setQ(A), which may or may not be wqo. A suitable concept covering this phenomenon is the notion of aQO-category. We have two conjectures aboutQO-categories in the effect that labellingQO-categories by a wqo set can always be reduced to labelling by ordinals. We prove these conjectures for a broad class ofQO-categories and for generalQO-categories we prove weaker forms of these conjectures.
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Kříž, I., Thomas, R. On well-quasi-ordering finite structures with labels. Graphs and Combinatorics 6, 41–49 (1990). https://doi.org/10.1007/BF01787479
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DOI: https://doi.org/10.1007/BF01787479