Supersolvable lattice

The definition encapsulates many of the nice properties of lattices of subgroups of supersolvable groups.

of elements (called an M-chain or chief chain) obeying any of the following equivalent properties.

For comparison, a finite lattice is geometric if and only if it is atomistic and the elements of the antichain of atoms are all left modular.

Working from the top, the chief chain is obtained by removing vertices in a perfect elimination ordering one by one.

[14][15] This is a consequence of a more general factorization theorem for characteristic polynomials over modular elements.

Any finite supersolvable lattice has an edge lexicographic labeling (or EL-labeling), hence its order complex is shellable and Cohen-Macaulay.

is supersolvable if and only if it has an edge lexicographic labeling that assigns to each maximal chain a permutation of