Finite metrics in switching classes

: Ehrenfeucht A, Harju T, Rozenberg G

Publisher: ELSEVIER SCIENCE BV

: 2007

Discrete Applied Mathematics

DISCRETE APPLIED MATHEMATICS

: DISCRETE APPL MATH

: 155

: 1

: 68

: 73

: 6

: 0166-218X

DOI: https://doi.org/10.1016/j.dam.2006.04.041

Lt g: D x D -> R be a symmetric function on a finite set D satisfying g(x, x) = 0 for all X E D. A switch g(n) of g w.r.t. a local valuation sigma: D -> R is defined by g(a)(x, y) = sigma(x) + g(x, y) + sigma(y) for x not equal y and g(sigma)(x, x) = 0 for all x. We show that every symmetric function g has a unique minimal semimetric switch, and, moreover, there is a switch of g that is isometric to a finite Manhattan metric. Also, for each metric on D, we associate an extension metric on the set of all nonempty subsets of D, and we show that this extended metric inherits the switching classes on D. (c) 2006 Elsevier B.V. All rights reserved.

Mathematics