Poset Edge-Labellings and Left Modularity
Abstract
It is known that a graded lattice of rank n is supersolvable if and only if it has an EL-labelling where the labels along any maximal chain are exactly the numbers 1,2,...,n without repetition. These labellings are called S_n EL-labellings, and having such a labelling is also equivalent to possessing a maximal chain of left modular elements. In the case of an ungraded lattice, there is a natural extension of S_n EL-labellings, called interpolating labellings. We show that admitting an interpolating labelling is again equivalent to possessing a maximal chain of left modular elements. Furthermore, we work in the setting of a general bounded poset as all the above results generalize to this case. We conclude by applying our results to show that the lattice of non-straddling partitions, which is not graded in general, has a maximal chain of left modular elements.
Cite
@article{arxiv.math/0211126,
title = {Poset Edge-Labellings and Left Modularity},
author = {Peter McNamara and Hugh Thomas},
journal= {arXiv preprint arXiv:math/0211126},
year = {2007}
}
Comments
16 pages, 5 figures. Some minor expository changes. To appear in the European Journal of Combinatorics. A new bonus section not included in the published version introduces the non-straddling partitions of [n], a subset of the non-nesting partitions of [n]. We show that the non-straddling partitions of [n] form a lattice possessing a maximal chain of left modular elements