An analogue of distributivity for ungraded lattices

Details An analogue of distributivity for ungraded lattices An analogue of distributivity for ungraded lattices

2005-02-14

arxiv.org/abs/math/0502278v2

Mathematics

Combinatorics

19 pages, 4 figures. Version 2 includes small improvements to exposition, corrections of typos, and a new section showing that

Scientific paper

In this paper, we define a property, trimness, for lattices. Trimness is a not-necessarily-graded generalization of distributivity; in particular, if a lattice is trim and graded, it is distributive. Trimness is preserved under taking intervals and suitable sublattices. Trim lattices satisfy a weakened form of modularity. The order complex of a trim lattice is contractible or homotopic to a sphere; the latter holds exactly if the maximum element of the lattice is a join of atoms. Other than distributive lattices, the main examples of trim lattices are the Tamari lattices and various generalizations of them. We show that the Cambrian lattices in types A and B defined by Reading are trim, and we conjecture that all Cambrian lattices are trim.

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