#### Title

Singular Points and an Upper Bound of Medians in Upper Semimodular Lattices

#### Document Type

Article

#### Publication Date

9-28-2000

#### 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.

#### Recommended Citation

Li, Jinlu and Boukaabar, Kaddour, "Singular Points and an Upper Bound of Medians in Upper Semimodular Lattices" (2000). *Faculty Emeritus*. 15.

https://digitalcommons.shawnee.edu/fac_emeritus/15