Order Based on Non-Associative Operations
Kavit Nanavati, Balasubramaniam Jayaram · Atlantis studies in uncertainty modelling/Atlantis Studies in Uncertainty Modelling · 2021
Recently there have been many works studying orders obtained from fuzzy logic connectives, using the relation proposed in [12].However, almost all of these works have dealt only with associative operations.In this work, we investigate the conditions under which the above relation leads to a partial order even when the operation is nonassociative, i.e., in the setting of a groupoid instead of a semigroup.We begin by presenting the necessary and sufficient conditions towards the same.Following this we study the major non-associative aggregation operations, viz., semi-copulas, fuzzy implications, overlapping and grouping functions.Our investigations show that even appropriate nonassociative functions can lead to interesting order-theoretic structures on the underlying set.