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.