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Proportional transitivity in linear extensions of ordered sets. (English) Zbl 0566.06002
Let $$p_{ij}$$ denote the proportion of all linear extensions $$>*$$ of a partial order on $$\{$$ 1,2,3,...,n$$\}$$ in which $$i>*j$$. The deterministic transitivity law for the subset $$\{$$ 1,2,3$$\}$$ says that $$(p_{12}=1,p_{23}=1)\Rightarrow p_{13}=1$$, and similarly for permutations of 123. The corresponding probabilistic or proportional transitivity law asserts that, for all ($$\lambda$$,$$\mu)$$ in the unit square, $$(p_{12}\geq \lambda,p_{23}\geq \mu)\Rightarrow p_{13}\geq f(\lambda,\mu)$$, where f($$\lambda$$,$$\mu)$$ is the infimum of $$p_{13}$$ over all finite posets that have $$p_{12}\geq \lambda$$ and $$p_{23}\geq \mu.$$
It is shown that $$f(\lambda,\mu)=0$$ when $$\lambda +\mu <1$$, and $$f(1,\mu)=\mu$$. Moreover, f($$\lambda$$,1-$$\lambda)\leq 1/e$$, and, when $$\lambda +\mu >1$$ and $$\max \{\lambda,\mu \}<1$$, f($$\lambda$$,$$\mu)\leq 1- (1-\lambda)(1-\mu)[1-\log (1-\lambda)(1-\mu)]$$. The exact value of f is presently unknown for every case in which $$\lambda +\mu \geq 1$$ and $$\max \{\lambda,\mu \}<1$$.

##### MSC:
 06A06 Partial orders, general
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##### References:
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