Abstract
The structure of uninorms which are continuous on some special parts of the unit square is discussed. After a summary of partial results achieved in the characterization of uninorms with continuous underlying t-norm and t-conorm in the past years, a full characterization of these uninorms is described. Representation theorems based on the set of discontinuity points of such a uninorm and the ordinal sum construction for semigroups are presented. Further generalizations yield uninorms with continuous diagonal functions. Several results related to uninorms with continuous diagonals are investigated. Further generalizations are also discussed.
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