The structure of rating scales

The structure of the set of all possible rating scales is investigated. It is shown that by a natural addition of rating scales the set is a commutative semigroup with neutral element. From this operation a partial order can be defined which turns out to a lattice order. This lattice is shown to be distributive. In the next step two possibilities-closely related to the preceding development-are analyzed to endow this structure with a metric. The semigroup operation is shown to be continuous in the respective topologies. With the help of one of these metrics the question of the scale type of rating scales is discussed by giving the concept of admissible transformations an extended meaning. (C) 2018 Elsevier Inc. All rights reserved.