A short note on the migrativity properties of overlap functions over uninorms

Abstract

Recently, as an addendum to Qiao and Hu (2018) [21], Zhu and Hu (2020) [25] studied the α-migrativity of overlap and grouping functions over uninorms and nullnorms. Later on, Zhou and Yan (2019) [24] further investigated the α-migrativity properties of overlap functions over uninorms. They showed equivalent characterizations of the (α,U)-migrativity equation when the uninorm U belongs to one certain class (e.g., Umin, Umax, the family of idempotent uninorms, representable uninorms or uninorms continuous on ]0,1[2). However, we find that Propositions 3.5, 3.8 and Remark 3.5 (2) presented by Zhu and Hu (2020) [25] are incorrect. In this note, at first, we establish the updated results. And then, we study the α-migrativity of overlap functions over t-norms when t-norms are continuous and give their characterizations. In particular, we point out that the relationship between the α-migrativity of overlap functions over uninorms and the α-migrativity of uninorms over overlap functions. Finally, we discuss the (α,U)-migrativity equation for grouping functions using an analogous method.