Balanced weightings and permutation matchings applied to the monoid of order-preserving mappings

Peter M. Higgins

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arXiv · 2610.01912

Balanced weightings and permutation matchings applied to the monoid of order-preserving mappings

Peter M. Higgins

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Abstract

We give a necessary and sufficient condition for a combinatorial D-class of a finite semigroup to have a permutation matching: that its structure matrix admit a balanced weighting, that is, non-negative weights with constant row sums and constant column sums. This weakens the regularity hypothesis, the role of Hall's lemma being taken up by the Birkhoff--von Neumann theorem. We then apply it to show that the monoid of all order-preserving self-maps of an n-element chain, has a permutation matching.

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Peter M. Higgins. 2026-10-01. Balanced weightings and permutation matchings applied to the monoid of order-preserving mappings. https://arxiv.org/abs/2610.01912

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