Properties of congruences of twisted partition monoids and their lattices

We build on the recent characterisation of congruences on the infinite twisted partition monoids P^Φ_n and their finite d-twisted homomorphic images P^Φ_n,d, and investigate their algebraic and order-theoretic properties. We prove that each congruence of P^Φ_n is (finitely) generated by at most ⌈5n/2⌉ pairs, and we characterise the principal ones. We also prove that the congruence lattice Cong(P^Φ_n) 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 Cong(P^Φ_n,d) is modular but still not distributive for d>0, while Cong(P^Φ_n,0) is distributive. We also calculate the number of congruences of P^Φ_n,d, showing that the array (|Cong(P^Φ_n,d)|)_n,d≥0 has a rational generating function, and that for a fixed n or d, |Cong(P^Φ_n,d)| is a polynomial in d or n≥4, respectively.