Pan-operations structure with non-idempotent pan-addition

The structure of pan-operations, involving a pan-addition and a pan-multiplication which form a commutative isotonic semiring, as defined by Wang and Klir, are discussed. Based on semi-group theory, special properties of identity element and the idempotent of pan-addition and pan-multiplication, the case of a non-idempotent pan-addition are studied. Our results correct a theorem of Mesiar and Rybarik. Moreover, a sufficient and necessary condition for the continuity of a pan-operator defined on [0,+∞] is presented. Our conclusions are expected to result in the introduction of the concept of general fuzzy integral.