Conditional distributivity equation of semi-uninorms over uninorms

The conditional distributivity of aggregation operators has always been the focus of research because it is crucial for various areas including integration theory, utility theory and so on. Inspired by this fact, this article is mainly devoted to dealing with the conditional distributivity of semi-uninorms over uninorms from Uemin∪Uemax, where semi-uninorms come from representable semi-uninorms and continuous semi-uninorms. The obtained results show that there is no representable semi-uninorm, which is conditionally distributive over such a uninorm. However, if the semi-uninorms are continuous, then the necessary and sufficient conditions for the conditional distributivity are given. And then based on the results we further show that distributivity and conditional distributivity are not equivalent in this case.