Poset Edge-Labellings and Left Modularity
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 Sn 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 Sn 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.
European Journal of Combinatorics
McNamara, Peter R. W. and Thomas, Hugh. "Poset Edge-Labellings and Left Modularity." European Journal of Combinatorics (2006) : 101-113.