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About k-optimal representations of posets. (English) Zbl 0681.06002

Author’s summary: “Let P be a finite partially ordered set. A function x: \(P\to {\mathbb{R}}\) is called a representation of P if \(x(p')-x(p)\geq 1\) whenever \(p'>p\). The representation x is said to be k-optimal if the k-th absolute central moment of x is minimal with respect to all other representations of P. The paper presents necessary and sufficient conditions for a representation to be k-optimal, shows that the rank function of normal posets, Peck posets, and distributive lattices is a k- optimal representation for all k, and proves that for every ranked poset with a unique minimal and a unique maximal element there exists a number \(k_ 0\) such that the rank function is a k-optimal representation for all \(k>k_ 0.''\)
Reviewer: J.Fronková

MSC:

06A06 Partial orders, general
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