PreprintCC0arXiv
Balanced weightings and permutation matchings applied to the monoid of order-preserving mappings
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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BibTeX
@misc{higgins2026balanced,
title = {{Balanced weightings and permutation matchings applied to the monoid of order-preserving mappings}},
author = {Peter M. Higgins},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01912},
eprint = {2610.01912},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01912},
note = {CC0},
}
RIS
TY - UNPB
TI - Balanced weightings and permutation matchings applied to the monoid of order-preserving mappings
AU - Peter M. Higgins
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01912
UR - https://arxiv.org/abs/2610.01912
AB - 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.
ER -