Characterization of uninorms with continuous underlying t-norm and t-conorm by means of the ordinal sum construction

The uninorms with continuous underlying t-norm and t-conorm are characterized via the ordinal sum construction of Clifford. Using the previous results of the author, where each uninorm with continuous underlying operations was characterized by properties of its set of discontinuity points, it is shown that each such a uninorm can be decomposed into an ordinal sum of a countable number of semigroups related to representable uninorms, continuous Archimedean t-norms, continuous Archimedean t-conorms and internal uninorms (including the min and the max operator).