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When Bjoerner and Wachs introduced one of the main forms of lexicographic shellability, namely CL-shellability, they also introduced the notion of recursive atom ordering, and they proved that a finite bounded poset is CL-shellable if and only if it admits a recursive atom ordering. We introduce a relaxation of the notion of recursive atom ordering, and we prove that any such relaxed recursive atom ordering may be transformed via a reordering process into a traditional recursive atom ordering. We use this to prove that several different notions of lexicographic shellability are all equivalent to each other, in the sense that any finite bounded poset admitting one of these admits all of them. As an application, we prove that the uncrossing orders, namely the face posets for stratified spaces of planar electrical networks, are dual CL-shellable. I will not assume familiarity with lexicographic shellability in this talk.

This is joint work with Grace Stadnyk.