Characterizing autodistributive aggregation operations defined on finite linearly ordered scales

Observe that for continuous, strictly increasing binary means defined on a unit interval, autodistributivity and bisymmetry are equivalent. Bisymmetric aggregation operations defined on finite linearly ordered scales have been studied in the past, and their behavior is unlike that of corresponding operations defined on the unit interval [0,1]. This paper investigates two families of autodistributive aggregation operations defined on finite linearly ordered scales: one with a neutral element and the other under some partial smoothness-related conditions. The former is fully described as the family of idempotent uninorms and the latter is completely characterized as the family of idempotent t-operators. Through these results, we can deduce that compared with bisymmetry, autodistributivity is much stronger, and the behavior of autodistributive aggregation operations defined on finite linearly ordered scales is significantly different from that of corresponding operations defined on [0,1].