Abstract
Given a k-tuple P=(x 1,x 2,...,x k ) in a finite lattice X endowed with the lattice metric d, a median of P is an element m of X minimizing the sum ∑ i d(m,x i ). If X is an upper semimodular lattice, Leclerc proved that a lower bound of the medians is c(P), the majority rule and he pointed out an open problem: “Is c 1(P)=∨ i x i , the upper bound of the medians?” This paper shows that the upper bound is not c 1(P) and gives the best possible upper bound.
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Li, J., Boukaabar, K. Singular Points and an Upper Bound of Medians in Upper Semimodular Lattices. Order 17, 287–299 (2000). https://doi.org/10.1023/A:1026760432073
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DOI: https://doi.org/10.1023/A:1026760432073