Decompositions of Augmented Bergman Complexes

The augmented Bergman complex of a closure operator on finite set interpolates between the order proper flats and independence operator. In 2020, Braden, Huh, Matherne, Proudfoot, Wang showed that complexes matroids are always gallery-connected, recently Bullock, Kelley, Reiner, Ren, Shemy, Shen, Sun, Tao, Zhang strengthened "gallery-connected" to "shellable" by providing two classes shelling orders: "flag-to-basis" shellings "basis-to-flag" shellings. 
 We show vertex decomposable, stronger property than shellable. also prove any is shellable if only its lattice (that is, non-augmented complex) As consequence, an it admits flag-to-basis shelling. Perhaps surprisingly, same does not hold for basis-to-flag shellings: we describe whose shellable, but has no with bases appearing first.