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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.