Properties of congruences of twisted partition monoids and their lattices

We build on the recent characterisation of congruences on the infinite twisted partition monoids (Formula presented.) and their finite (Formula presented.) -twisted homomorphic images (Formula presented.), and investigate their algebraic and order-theoretic properties. We prove that each congruence of (Formula presented.) is (finitely) generated by at most (Formula presented.) pairs, and we characterise the principal ones. We also prove that the congruence lattice (Formula presented.) is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending chains and infinite anti-chains. By way of contrast, the lattice (Formula presented.) is modular but still not distributive for (Formula presented.), while (Formula presented.) is distributive. We also calculate the number of congruences of (Formula presented.), showing that the array (Formula presented.) has a rational generating function, and that for a fixed (Formula presented.) or (Formula presented.), (Formula presented.) is a polynomial in (Formula presented.) or (Formula presented.), respectively.