Coherence and avoidance of sure loss for standardized functions and semicopulas

We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value 1 at (1, 1, ... , 1). We characterize the existence of a k-increasing n-variate function C fulfilling A <= C <= B for standardized n-variate functions A, B and discuss methods for constructing such functions. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when A respectively B coincides with the pointwise infimum respectively supremum of the set of all k-increasing n-variate functions C fulfilling A <= C <= B.