Metric Properties of Tolerances
Gérard Kientega and Ivo Rosenberg
Let A be a nonindexed algebra with base set E. We define a generalized metric with values in the ordered monoid of the tolerances of A. The binary law of the monoid is the composition of relations and the order that we consider is induced by the lattice structure of the tolerances of A.We then study the properties of this metric and use them to characterize tolerance affine complete and locally tolerance affine complete algebras. We hence improve certain results of Chajda. Those new ideas allow us to establish all our theorems for a single algebra. Usually an accurate description of affine complete or tolerance affine complete algebras can only be made by considering the whole variety.
Keywords: Generalized Metric, Helly property, binary reflexive relation, polynomial, tolerance affine complete algebra, local clone.
MSC 2000: 08A40, 08A30, 54E35.