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

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.