Pattern Recognition on Oriented Matroids

5.6The Blocker, Deletion, Contraction, and Maps on Posets

In this section we give yet another poset-theoretic interpretation of the fundamental identities (5.3) which establish the connection between the set-theoretic blocker, deletion, and contraction maps on clutters.

Theorem 5.25. Let L be a finite poset. Let δ, γ : L → L be order-preserving maps on L such that

$\gamma \left(x\right)\underset{\_}{\succ }x,\left(5.22\right)$

for all x ∈ L.

Let β · L → L be an order-reversing map such that

$\beta \left(\beta \left(x\right)\right)\underset{\_}{\succ }x,\left(5.23\right)$

for all x ∈ L.

If the relations

$\beta \left(\delta \left(\beta \left(x\right)\right)\right)\underset{\_}{\succ }\gamma \left(x\right)\left(5.24\right)$

and

$\beta \left(\gamma \left(\beta \left(x\right)\right)\right)\underset{\_}{\succ }\delta \left(x\right)\left(5.25\right)$

are valid for all x ∈ L, then ...