The \(\alpha \) -cross-migrativity as a particularly interesting and important feature of binary operators has been researched in many literatures. In particular, Zhu et al. (Fuzzy Sets Syst 451:113–129, 2022b) studied the \(\alpha \) -cross-migrativity between uninorms (nullnorms) and overlap (grouping) functions. It is also worth noting that uni-nullnorms (null-uninorms) are the generalization of uninorms and nullnorms. Based on this considerations, it is necessary to further study the \(\alpha \) -cross-migrativity between uni-nullnorms (null-uninorms) and overlap (grouping) functions. In this paper, we conduct a detailed investigation into the \(\alpha \) -cross-migrativity between proper conjunctive uni-nullnorms (with continuous Archimedean underlying t-norms and t-conorms) and overlap functions by partitioning the value range of \(\alpha \) . By utilizing the ordinal sums of t-norms, we obtain some equivalence characterizations on the \(\alpha \) -cross-migrativity equations. Further, by means of the additive generators of overlap (grouping) functions, some equivalent conditions are transformed into deeper conclusions. Finally, based on the duality, the \(\alpha \) -cross-migrativity between proper null-uninorms and grouping (overlap) functions is studied in a similar way.
