Characterizing set-valued functions of uninorms with continuous underlying t-norm and t-conorm

The uninorms with continuous underlying functions were characterized by their set of discontinuity points in the previous work of the author, using the characterizing set-valued function. In this paper properties of this characterizing set-valued function are studied. It is shown that the type of the monotonicity of such a set-valued function is always changed in idempotent points of the corresponding uninorm. Several additional properties of the characterizing set-valued function of a uninorm with continuous underlying functions are shown and an example is included. Results shown in this paper are used for a complete characterization of uninorms with continuous underlying functions via the ordinal sum construction, which is shown in the consecutive paper.