Uninorms internal on one or more non-trivial cuts

Abstract

Each uninorm is trivially internal on the cut in the neutral element and uninorms internal on the boundary were already completely characterized. We discuss uninorms which are internal on some non-trivial cut and show that each such uninorm can be expressed as a non-trivial ordinal sum. In individual cases we discuss the structure of such conjunctive (disjunctive) uninorm and show that the corresponding ordinal sum contains uninorms, t-conorms, t-superconorms and generalized sub-uninorms (t-norms, t-subnorms and generalized super-uninorms). Moreover, we discuss the ordinal sum decomposition of uninorms with several internal cuts and show their decomposition into semigroups that have only constant internal cuts, and possibly an internal cut in the neutral element.