Poset edge-labellings and left modularity

It is known that a graded lattice of rank nn is supersolvable if and only if it has an EL-labelling where the labels along any maximal chain are exactly the numbers 1,2,…,n1,2,…,n without repetition. These labellings are called SnSn 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 SnSn 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 an arbitrary bounded poset as all the above results generalize to this case.