On coarsenings of L-sets

Usually, L-sets are understood to be functions an the universe U, where L is a complete lattice satisfying, perhaps some additional properties. However knowledge about the universe, e.g. topologies, may restrict us to determine level sets of L-sets. L-sets are defined on a topological space and in this case some level sets may not be closed sets in that topology. Then, Representation Theorems, presented by Negoita, and Ralescu (1993), can be applied to determine a class of coarsenings of an L-set defined by means of a topology on U. An operation for computing a specific coarsening of an L-set is proposed and some properties of this operation is studied.