Singular Points and an Upper Bound of Medians in Upper Semimodular Lattices
Original Publication Title
Given a metric space (X, d) and a k-tuple (profile) P = (x1, x2,...,xk ) of elements of P X, a median is an element m of X minimizing the remoteness r(m) = i d(m, xi). In a special case that X is a finite upper semimodular lattice, Leclerc proved  that a lower bound of the medians is c(P ), the majority rule. By duality, an upper bound of the medians is c0 (P ), the dual majority rule, if X is a lower semimodular lattice. Both c(P ) and c0 (P ) will be defined in Section 2. These imply that in a finite modular lattice, the upper and the lower bounds of the medians will be c0 (P ) and c(P ), respectively. In the same paper, Leclerc pointed out an open problem: what is an upper bound for the medians in an upper semimodular lattice? Is c1(P ) (= W i xi) the upper bound of the medians? This paper shows that the upper bound is not c1(P ) by some examples (in Section 3) and gives the best possible upper bound (in Section 4).
Li, Jinlu, "Singular Points and an Upper Bound of Medians in Upper Semimodular Lattices" (2000). Faculty Research. 12.