A presentation for the monoid of uniform block permutations

Abstract

The monoid Fn of uniform block bijections is the factorizable inverse monoid which arises from the natural action of the symmetric group on the join semilattice of equivalences on an n-set; it has been described in the literature as the factorizable part of the dual symmetric inverse monoid. The present paper gives and proves correct a monoid presentation for Fn: The methods involved make use of a general criterion for a monoid generated by a group and an idempotent to be inverse, the structure of factorizable inverse monoids, and presentations of the symmetric group and the join semilattice of equivalences on an n-set.