Upper bounding overlaps by groupings

The notions of overlapping and grouping arise from the problem how to assign an object to exactly one class among several available. In this respect, overlaps and groupings are connected with t-norms and t-conorms. However, there are some properties of t-norms and t-conorms that are not valid for overlaps and groupings, for example, that any t-norm is less than or equal than any t-conorm. This fact motivates the proposed research. Specifically, we determine conditions to ensure that an overlap is smaller than a certain grouping and methods to define operators satisfying such ordering. The relevance of this study is visible at the end of the paper by the inclusion of an illustrative example in the context of image processing.