Pattern Recognition on Oriented Matroids

6.5Principal Order Ideals and Farey Subsequences

In this section we define Farey subsequences which can be useful for the existence problems on the relatively blocking elements in posets.

Let {a} be a nontrivial one-element antichain of a finite bounded poset P. Consider the sequence of irreducible fractions

$\begin{array}{l}F\left(P,a;\omega \right):=\left\{\frac{0}{1},\frac{1}{1}\right\}\\ \cup \left(\frac{\omega \left(\left\{b\right\}{\wedge }_{\Delta }\left\{a\right\}\right)}{\gcd \left(\omega \left\{b\right\}{\wedge }_{\Delta }\left\{a\right\},\omega \left(b\right)\right)}/\frac{\omega \left(b\right)}{\gcd \left(\omega \left(\left\{b\right\}{\wedge }_{\Delta }\left\{a\right\}\right),\omega \left(b\right)\right)}:b\in P-\left\{{\text{<mover accent="true"> O <mo>^</mo> </mover>}}_P\right\}\right)\left(6.24\right)\end{array}$

arranged in increasing order. Note that (P, a; ω) is a subsequence of the Farey sequence of order ω(P).

We always ...