Graded dominance

The relation of dominance between aggregation operators has recently been studied quite intensively [9, 12, 10, 11, 13, 14].We propose to study its 'graded' generalization in the foundational framework of higher-order fuzzy logic, also known as Fuzzy Class Theory (FCT) introduced in [1]. FCT is specially designed to allow a quick and sound development of graded, lattice-valued generalizations of the notions of traditional 'fuzzy mathematics' and is a backbone of a broader program of logic-based foundations for fuzzy mathematics, described in [2]. This short abstract is to be understood as just a 'teaser' of the broad and potentially very interesting area of graded dominance. We sketch basic definitions and properties related to this notion and present a few examples of results in the area of equivalence and order relations (in particular, we show interesting graded generalization of basic results from [6, 12]). Also some of our theorems are, for expository purposes, stated in a less general form here and can be further generalized substantively. In this paper, we work in Fuzzy Class Theory over the logic MTLD of all left-continuous t-norms [7]. The apparatus of FCT and its standard notation is explained in detail in the primer [3], which is freely available online. Furthermore we use X vY for D(X vY).