Pattern Recognition on Oriented Matroids by Andrey O. Matveev

Author:Andrey O. Matveev [Matveev, Andrey O.]
Language: eng
Format: mobi, pdf
Publisher: De Gruyter
Published: 2017-07-26T21:00:00+00:00

Hence, eq. (5.43) follows.

We conclude this section by applying standard techniques from the poset theory to the lattice (P) and to the (X, k)-blocker map.

Proposition 5.34. Let Pbea finite bounded poset.

(i) The composite map

is a closure operator on the lattice of antichains (P).

(ii) The poset is a self-dual lattice; the restriction map is an anti-automorphism of is a sub-meet-semilattice of the lattice (P).

(iii) For any its preimage under the (X, k)-blocker map is a convex sub-join-semilattice of the lattice (P). The greatest element of is

Proof. In view of Lemma 5.32(ii) and Lemma 5.33, the assertions (i) and (ii) follow from the well-known results on closure operators.

To prove assertion (iii), pick arbitrary elements where for some A ∈ (P), and note that If is the greatest element of is the one-element subposet Finally, if B is a nontrivial antichain of P, then by eq. (5.44) the element is the greatest element of Since the (X, k)-blocker map is order-reversing, we see that the subposet of (P) is convex.

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