Pattern Recognition on Oriented Matroids

5.8The (X, k)-Blocker Map

In this section we consider a family of maps on the antichains of a finite bounded poset that generalize the set-theoretic blocker map on clutters, as well as the poset-theoretic blocker map on antichains, defined by eq. (5.11).

Let X ⊆ Pa be a subset of the set of atoms of a finite bounded poset P, and k a nonnegative integer, k ≤ |Pa|.

Definition 5.29.

(i)The (X, k)-blocker map on the lattice of antichains of a finite bounded poset P is the map ${\text{b}}_k^X:\mathfrak{A}\left(P\right)\to \mathfrak{A}\left(P\right)$ given by

$\text{A}\mapsto min\left\{b\in P:\left|\mathfrak{J}\left(b\right)\cap \mathfrak{J}\left(a\right)\cap \left(P^{\text{a}}-X\right)\right|>k,\forall a\in \text{A}\right\}$

if the antichain A is nontrivial, and

${\tilde{0}}_{\mathfrak{A}\left(P\right)}\mapsto {\tilde{1}}_{\mathfrak{A}\left(P\right)},{\tilde{1}}_{\mathfrak{A}\left(P\right)}\mapsto {\tilde{0}}_{\mathfrak{A}\left(P\right)}.$

(ii)Given an antichain A ∈ (P), the (X, k)-blocker of A in P is the ...