Averaging procedures in defuzzification processes

Defuzzification is the ultimate step in approximate reasoning, consisting in the replacement of a fuzzy set with a suitable crisp value. This process is decomposed in two steps: first, replacing a fuzzy set by a crisp set, then replacing the obtained crisp set by a single value. We investigate the natural conditions the first replacement--called averaging procedure--should satisfy. Some interesting examples are given. The compatibility of averaging procedures with algebraic and order structures is studied. The algebraic structure of the set of averaging procedures is investigated.