Distributivity and conditional distributivity of semi-uninorms over continuous t-conorms and t-norms.

The distributivity and conditional distributivity of a uninorm and a continuous t-conorm present an open problem recalled by Klement in the Linz2000 closing session. In 2006, Ruiz and Torrens 14 solved this problem for the most usual known classes of uninorms. Recently, Rak 13 solved the same problem for several classes of semi-uninorms, i.e., a special class of conjunctive semi-uninorms, a special class of disjunctive semi-uninorms and the class of idempotent semi-uninorms. In this work we continue to investigate the same topic as the above by focusing on two other classes of semi-uninorms, i.e., representable semi-uninorms and continuous semi-uninorms. The obtained results are different from the case of representable uninorms and also different from the cases of other known classes of semi-uninorms. Moreover, the dual case of distributivity and conditional distributivity of the above two classes of semi-uninorms over continuous t-norms is also investigated and similar results are obtained.