Pattern Recognition on Oriented Matroids

6.3A Connection Between Absolute and Relative Blocking Constructions

It follows from Definition 6.5 that the relative 0-blocker 0(A) of a nontrivial antichain A in P, with respect to an arbitrary map ω, is nothing else than the absolute 0-blocker $\mathfrak{b}\left(A\right):={\mathfrak{b}}_0\left(A\right):={\mathfrak{b}}_0^{{\cancel{0}}^a}\left(A\right),$ or just the blocker for short, of A in P, considered in Chapter 5. Moreover, if $\mathfrak{b}\left(A\right)\subseteq P^1,\text{then }{\cap }_{a\in A}\mathfrak{J}\left(a\right)-\left\{{\text{<mover accent="true"> O <mo>⌢</mo> </mover>}}_P\right\}\subseteq I_r\left(P,A;\omega \right),$ and r(A) = (A), for any value of the parameter r.

Again let A be a nontrivial antichain of P, and let ...