Properties of congruences of twisted partition monoids and their lattices
James Edward East, Nik Ruškuc · arXiv (Cornell University) · 2020
We build on the recent characterisation of congruences on the infinite twisted partition monoids $\mathcal{P}_{n}^Φ$ and their finite $d$-twisted homomorphic images $\mathcal{P}_{n,d}^Φ$, and investigate their algebraic and order-theoretic properties. We prove that each congruence of $\mathcal{P}_{n}^Φ$ 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 $\textsf{Cong}(\mathcal{P}_{n}^Φ)$ 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 $\textsf{Cong}(\mathcal{P}_{n,d}^Φ)$ is modular but still not distributive for $d>0$, while $\textsf{Cong}(\mathcal{P}_{n,0}^Φ)$ is distributive. We also calculate the number of congruences of $\mathcal{P}_{n,d}^Φ$, showing that the array $\big(|\textsf{Cong}(\mathcal{P}_{n,d}^Φ)|\big)_{n,d\geq 0}$ has a rational generating function, and that for a fixed $n$ or $d$, $|\textsf{Cong}(\mathcal{P}_{n,d}^Φ)|$ is a polynomial in $d$ or $n\geq 4$, respectively.