Presentations for rook partition monoids and algebras and their singular ideals

TL;DR

This paper provides new algebraic presentations for rook partition monoids and algebras, analyzes their singular ideals, and determines minimal generating sets, highlighting the role of idempotents in generation.

Contribution

It introduces new presentations with minimal generating sets and characterizes the singular ideals of rook partition monoids and algebras.

Findings

Presented generators and relations for rook partition monoids and algebras.

Calculated minimal sizes of generating sets.

Showed singular part is generated by idempotents.

Abstract

We obtain several presentations by generators and relations for the rook partition monoids and algebras, as well as their singular ideals. Among other results, we also calculate the minimal sizes of generating sets (some of our presentations use such minimal-size generating sets), and show that the singular part of the rook partition monoid is generated by its idempotents.