On the coverings by tolerance classes

A tolerance is a reflexive and symmetric, but not necessarily transitive, binary relation. Contrary what happens with equivalence relations, when dealing with tolerances one must distinguish between blocks (maximal subsets where the tolerance is a total relation) and classes (the class of an element is the set of those elements tolerable with it). Both blocks and classes of a tolerance on a set define coverings of this set, but not every covering of a set is defined in this way. The characterization of those coverings that are families of blocks of some tolerance has been known for more than a decade now. In this paper we give a characterization of those coverings of a finite set that are families of classes of some tolerance.