Properties of congruences of twisted partition monoids and their lattices

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