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'Leave no one behind' has become an important principle of the entire 2030 Sustainable Development Agenda and the Sustainable Development Goals that brings inequality to the centre stage. Although a precise definition of what it is to lag behind is essential to assess sustainable economic growth and social progress a discussion on the desirable properties of such a measure is absent in the literature. This paper fills that gap by proposing and discussing a number of normative and analytical properties a measure of the 'Leave no one behind' principle should satisfy. The axioms proposed are necessary and sufficient to characterize the fuzzy measure by García-Pardo et al. [8] and together with other additional axioms make the structure of this measure more transparent. • Leaving no one behind is the central focus of the Sustainable Development Goals. • The Leaving no one behind principle suggests going beyond the averages. • The ambiguity of the left behind principle makes it convenient to use a fuzzy approach. • We discuss desirable properties of a measure of the extent an individual lags behind. • A subset of axioms characterizes the fuzzy set proposed by García-Pardo et al. [8]. [ABSTRACT FROM AUTHOR]

In this paper we present a fuzzy multidimensional approach for the measurement of the 'leaving no one behind' principle underlying the Sustainable Development Goals (SDGs) in European countries. In particular, we consider a two-step procedure. First, we compute the degree to which an individual is 'left behind' in each specific dimension. We then propose alternatives to measure the extent to which an individual is 'left behind' in a multidimensional setting. We illustrate our proposal taking as a reference the 'at risk of poverty or social exclusion' (AROPE) framework (dimensions, indicators and union criteria). The results highlight that although the average 'left behind' (LB) level and AROPE are largely consistent across European countries, our fuzzy measure provides valuable additional information, both in individual and aggregate terms, on persons who are further away from those better positioned and how far they are. [ABSTRACT FROM AUTHOR]

In this work, we focus on adjunctions, also called isotone Galois connections, in the framework of fuzzy preordered sets (hereafter, fuzzy preposets). Specifically, we present necessary and sufficient conditions so that, given a mapping f : A → B from a fuzzy preposet A into an unstructured set B , it is possible to construct a suitable fuzzy preorder relation on B for which there exists a mapping g : B → A such that the pair ( f , g ) constitutes an adjunction. [ABSTRACT FROM AUTHOR]