Presentations for semigroups of full-domain partitions

TL;DR

This paper provides algebraic presentations for the full-domain partition monoid, its singular ideal, and its planar submonoid, revealing their structural properties and connections to diagram monoids and category theory.

Contribution

It offers the first generators-and-relations presentations for these monoids, including the planar submonoid, and clarifies their algebraic structure and relationships.

Findings

Presented generators and relations for $P_n^{fd}$ and its submonoids.

Identified the structure of the planar submonoid as a grrac monoid.

Connected the monoid structures to diagram monoids and category theory concepts.

Abstract

The full-domain partition monoid $P_n^{fd}$ has been discovered independently in two recent studies on connections between diagram monoids and category theory. It is a right restriction Ehresmann monoid, and contains both the full transformation monoid and the join semilattice of equivalence relations. In this paper we give presentations (by generators and relations) for $P_n^{fd}$, its singular ideal, and its planar submonoid. The latter is not an Ehresmann submonoid, but it is a so-called grrac monoid in the terminology of Branco, Gomes and Gould. In particular, its structure is determined in part by a right regular band in one-one correspondence with planar equivalences.