Pattern Recognition on Oriented Matroids

6.9Relatively Blocking Elements in the Posets Isomorphic to the Face Lattices of Crosspolytopes

In this section we are concerned with the enumeration of the relatively blocking elements in the posets isomorphic to the face lattices ofcrosspolytopes.

Consider a poset O′(m) isomorphic to the face meet-semilattice of the boundary of an m-dimensional crosspolytope, which is defined as follows: the semilattice O′(m) consists of all the subsets of the set ±[m], free of pairs of opposites {–i, i}, and ordered by inclusion. We define a rank (m + 1) lattice O(m), with the rank function ρ(·), isomorphic to the face lattice of an m-dimensional crosspoly tope by

$\text{O}\left(m\right):\text{<msup> O <mo>′</mo> </msup>}\left(m\right)\cup \{\text{f},}$

where is a greatest element adjoined.

Let a subposet A ⊂ O(m)–{ ...