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Additive Generator Pairs of Overlap and Grouping Functions in Fuzzy Aggregation

How overlap and grouping functions can be constructed from pairs of one-variable generator functions, and what relationship those generators must satisfy. The construction O(x,y)=vartheta(theta(x)+theta(y)) reduces a bivariate aggregation operator to univariate functions; this characterizes when such an additively generated overlap function is a triangular norm and when it is a distortion of a t-norm by a pseudo automorphism, with dual results for grouping functions.

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Note on additive generator pairs of overlap and grouping functions∗ Li-zhi Liang†, Xue-ping Wang‡,

A fuzzy-set-theory paper on overlap and grouping functions, which are commutative, increasing, continuous binary aggregation operators on the unit square used for classification and decision-making w…