Survey on associative aggregation functions continuous around diagonal

This paper is the final step in the characterization of associative aggregation functions continuous around the diagonal. First, we survey all previous results concerning both commutative and non-commutative associative aggregation functions. Then we extend the result which shows that all commutative associative aggregation functions continuous around the diagonal can be expressed as a z -ordinal sum of semigroups with continuous additive generators, and show that also without commutativity, all non-commutative associative aggregation functions continuous around the diagonal can be expressed as a non-commutative ordinal sum of semigroups with continuous additive generators. Therefore, all associative aggregation functions continuous around the diagonal are characterized by inter and intra aggregation with respect to the corresponding summands.