Title: A Natural Classification Of Isotone Real Mappings
����� Let �S� be a semigroup of mappings defined on an interval �I� of the reals. Assume �S� contains all order automorphisms of �I�. The members of �S� can then be classified by how they act on arbitrary �k�-tuples from �I�. The resulting classifications will be exhibited in concrete cases, and the results applied to define corresponding classifications of cluster functions. New insights into the subject will allow a clearer exposition that will unify and clarify earlier results. These results involve isotone mappings on the reals, 0-preserving isotone mappings on the nonnegative reals, residuated mappings on the nonnegative reals, and residuated mappings on [0,1] having 1 in their image.