Mathematics – Combinatorics
Scientific paper
2002-11-07
European Journal of Combinatorics 27 (1) (2006), 101-113
Mathematics
Combinatorics
16 pages, 5 figures. Some minor expository changes. To appear in the European Journal of Combinatorics. A new bonus section no
Scientific paper
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.
McNamara Peter
Thomas Helmuth
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