Supersolvable Antimatroids
Motivated by a problem of defining partial orders on the elements of a Coxeter group, Armstrong (2007) studied antimatroids which are also supersolvable lattices. A supersolvable antimatroid is defined by a totally ordered collection of elements, and a family of sets of these elements. The family must include the empty set. Additionally, it must have the property that if two sets A and B belong to the family, the set-theoretic difference B \ A is nonempty, and x is the smallest element of B \ A, then A ∪ {x} also belongs to the family. As Armstrong observes, any family of sets of this type forms an antimatroid. Armstrong also provides a lattice-theoretic characterization of the antimatroids that this construction can form.
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