General ordinal sums of (pseudo-quasi) overlap and grouping functions

Abstract

In this work, we extend the existing ordinal sum constructions of fuzzy negations and fuzzy Sheffer strokes, which are based on the standard negation, to general ordinal sum structures derived from arbitrary negations. We also establish equivalent characterizations of their fundamental properties. Building upon this foundation and leveraging the completeness of the fuzzy Sheffer stroke, which provides a robust theoretical basis for designing fuzzy logic operators, we construct general ordinal sum structures for overlap (grouping) functions and pseudo-quasi overlap (grouping) functions. In these novel constructions, the range of function values within each summand interval is extended from the summand interval itself to an interval determined jointly by the summand interval and an arbitrary automorphism or quasi automorphism. Similarly, the function values outside the summand intervals are generalized from the identity mapping to an automorphism or quasi automorphism on [0,1]. This approach not only generalizes the existing corresponding ordinal sum structures but also substantially broadens their scope, thereby offering new perspectives for developing richer and more flexible ordinal sum constructions for other fuzzy logic operators.