Conditions of modularity of the congruence lattice of an act over a rectangular band

We describe polygons over rectangular bands that have modular, distributive or linearly ordered congruence lattice. It turns out that such polygons have at most 11 elements, and their congruence lattice has at most 300 elements. Furthermore, certain facts are established about the structure of polygons with modular congruence lattice over an arbitrary semigroup and about the structure of the congruence lattice of a polygon over a rectangular band. The work is based on the description of polygons over a completely (0-)simple semigroup obtained by Avdeev and Kozhukhov in 2000 and on the characterization of disconnected polygons with modular or distributive congruence lattice by Ptakhov and Stepanova in 2013.