On the partition monoid and some related semigroups

TLDR: The partition monoid is a salient natural example of a *-regular semigroup, and it has been shown that it contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set.

Abstract: The partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.