Distances in finite spaces from noncommutative geometry

Following the general principles of nuncommutative geometry, it is possible to define a metric on the space of pure slates of the noncommutative algeb ra generated by the coordinates. This metric generalizes the usual Riemanni an one. We investigate some general properties of this metric in finite com mutative cases corresponding to a metric on a finite set, and also compute explicitly some distances associated to commutative or noncommutative algeb ras. (C) 2001 Elsevier Science B.V. All rights reserved. MSC: 46L87.