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The median procedure in the semilattice of orders. (English) Zbl 1026.06006
Summary: Let $X$ be a finite set; we are concerned with the problem of finding a consensus order $P$ that summarizes an $m$-tuple (profile) $P^*$ of (partial) orders on $X$. A classical approach is to consider a distance function $d$ on the set $O$ of all the orders of $X$ and to search to minimize the remoteness $\sum_{1\le i\le m} d(P,P_i)$. We study some properties of this median procedure, and compare it with some other consensus approaches. Besides the classical symmetric difference metric, other distances are considered, and we particularly address the consequences for the consensus problem of the existence of a semilattice structure on the set $O$.

##### MSC:
 06A12 Semilattices 91B14 Social choice 06A06 Partial order
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