Properties of congruences of twisted partition monoids and their lattices

Abstract

We build on the recent characterisation of congruences on the infinite twisted partition monoids ${\mathcal{P}}_n^{\Phi }$ and their finite $d$-twisted homomorphic images ${\mathcal{P}}_{n,d}^{\Phi }$, and investigate their algebraic and order-theoretic properties. We prove that each congruence of ${\mathcal{P}}_n^{\Phi }$ is (finitely) generated by at most $\lceil \frac{5n}{2}\rceil$ pairs, and we characterise the principal ones. We also prove that the congruence lattice $\text{Cong}({\mathcal{P}}_n^{\Phi })$ is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending chains and infinite antichains. By way of contrast, the lattice $\text{Cong}({\mathcal{P}}_{n,d}^{\Phi })$ is modular but still not distributive for $d>0$, while $\text{Cong}({\mathcal{P}}_{n,0}^{\Phi })$ is distributive. We also calculate the number of congruences of ${\mathcal{P}}_{n,d}^{\Phi }$, showing that the array $(|\text{Cong}({\mathcal{P}}_{n,d}^{\Phi })|{)}_{n,d\ge 0}$ has a rational generating function, and that for a fixed $n$ or $d$, $|\text{Cong}({\mathcal{P}}_{n,d}^{\Phi })|$ is a polynomial in $d$ or $n\ge 4$, respectively.

41 pages, 8 figures, 4 tables, to appear in the J. London Math. Soc