Order congruence lattices are EL-shellable

Russ Woodroofe
Mississippi State University

Jay Schweig
Oklahoma State University

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Minisymposium: COMBINATORICS

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.

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