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.