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We introduce the concept of shuffle- and union-compatibility of total orders on words over a finite alphabet. This works only in a graded manner respecting the length of words. We classify all such orders by even exhibiting a normal form. For union-compatibility we draw on experience gained in a generalisation of Kruskal's packing theorem, as well as in working with rigidified surjections, for shuffle compatibility we also draw on Saizeva-Trevisan's characterisation of compatible total orders on finitely generated free commutative groups.
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Leeb, K., Pirillo, G. Shuffle-compatible total orders. Annali di Matematica pura ed applicata 153, 1–26 (1988). https://doi.org/10.1007/BF01762383
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DOI: https://doi.org/10.1007/BF01762383