Overlap functions are a class of aggregation functions that measure the overlapping degree between two values. Interval-valued overlap functions were defined as an extension to express the overlapping of interval-valued data, and they have been usually applied when there is uncertainty regarding the assignment of membership degrees. The choice of a total order for intervals can be significant, which motivated the recent developments on interval-valued aggregation functions and interval-valued overlap functions that are increasing to a given admissible order, that is, a total order that refines the usual partial order for intervals. Also, width preservation has been considered on these recent works, in an intent to avoid the uncertainty increase and guarantee the information quality, but no deeper study was made regarding the relation between the widths of the input intervals and the output interval, when applying interval-valued functions, or how one can control such uncertainty propagation based on this relation. Thus, in this paper we: (i) introduce and develop the concepts of width-limited interval-valued functions and width limiting functions, presenting a theoretical approach to analyze the relation between the widths of the input and output intervals of bivariate interval-valued functions, with special attention to interval-valued aggregation functions; (ii) introduce the concept of $(a,b)$-ultramodular aggregation functions, a less restrictive extension of one-dimension convexity for bivariate aggregation functions, which have an important predictable behaviour with respect to the width when extended to the interval-valued context; (iii) define width-limited interval-valued overlap functions, taking into account a function that controls the width of the output interval; (iv) present and compare three construction methods for these width-limited interval-valued overlap functions.

In this work we extend to the interval-valued setting the notion of an overlap functions and we discuss a method which makes use of interval-valued overlap functions for constructing OWA operators with interval-valued weights. . Some properties of intervalvalued overlap functions and the derived interval-valued OWA operators are analysed. We specially focus on the homogeneity and migrativity properties. Keywords Interval-valued fuzzy sets interval-valued overlap functions Interval-valued overlap OWA operators interval weighted vector migrativity homogeneity 1 Introduction Interval-valued fuzzy sets [62] have been succesfully applied in many different problems. Just to mention some of the most recent ones, interval-valued fuzzy sets have been used in decision making(see, e.g., theworksbyKhalilandHassan[36]andChengetal. They have also been the origin of rich theoretical studies, as, for instance, the works by Bedregal et al. [3, 7], Dimuro et al. [28], Reiser et al. [48] and the recent works by Zywica et al. [64] and Takác [55]. From the point of view of applications, interval-valued fuzzy sets are a suitable tool to represent uncertain or incomplete information. In particular, the length of the intervalvalued membership degree of a given element can be understood as a measure of the lack of certainty of the expert for providing an exact membership value to that element [44].