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On proportional transitivity of ordered sets. (English) Zbl 0912.06005
Summary: For a finite partially ordered set (poset) $$P$$, denote by $$p (x<y)$$ the fraction of linear extensions of $$P$$ in which $$x$$ precedes $$y$$. It is shown that if $$p (x<y)$$ and $$p (y< z)$$ are each at least $$u$$ with $$u\geq \rho\approx 0.78$$, then $$p (x<z)$$ is at least $$u$$. The result stated is mainly a consequence of the $$XYZ$$ inequality [L. A. Shepp, Ann. Probab. 10, 824-827 (1982; Zbl 0484.60010)] and a theorem of K. Ball [Stud. Math. 88, 69-84 (1988; Zbl 0642.52011)] which allows us to reduce to a two-dimensional version of the problem.

MSC:
 06A07 Combinatorics of partially ordered sets
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