Order congruence lattices are EL-shellable
Mississippi State University
Oklahoma State University
Content: The lattice of order congruences of a poset P is the subposet O(P) of the lattice of partitions of P, consisting of all partitions that satisfy a certain order-convexity property. The partition lattice is both semi-modular and supersolvable, but Körtesi, Radelecki and Szilágyi gave examples of order congruence lattices that are not semi-modular, and we find examples that are not supersolvable. We show that the order congruence lattices satisfy a recursive condition on the existence of modular elements, which we call lm-decomposability. Lm-decomposability generalizes both supersolvability and semimodularity. We show that any lm-decomposable lattice is EL-shellable. This result simultaneously improves a weaker result of Jenča and Sarkoci, and also exposes inclusion-exclusion information (Möbius numbers). This project is joint with Jay Schweig.