A unified representation of intuitionistic fuzzy sets, hesitant fuzzy sets and generalized hesitant fuzzy sets based on their u-maps

Since its creation by Zadeh in 1965, fuzzy set theory (FST) has been continuously advanced in various fronts during the past five decades. Along with Zadeh's classical FST (also termed type-1 fuzzy set theory), a number of higher-order FSTs, including e.g. type-2 fuzzy sets, intuitionistic fuzzy sets, typical hesitant fuzzy sets, and generalized hesitant fuzzy sets, have been proposed, constructed, and applied. A quick survey of the literature leads one to observe that a large amount of remarkable researches has been performed on developing theories of these higher-order fuzzy sets - a significant portion of these researches have involved painstaking efforts in devising suitable fundamental operators (e.g. the basic set operators and the aggregation operators) and measures (e.g. the similarity, subsethood, and entropy measures) under these various higher-order settings. At the same time, one also observes that the somewhat disparate frameworks assumed under these various higher-order settings have led to a highly complex landscape of the whole FST field. Arguably, this complexity poses significant barriers for any non-expert to try to apply these latest developments to his/her application domain. In this article, based on a so-called u-map representation that we have developed, we propose a very simple framework, via suitably adapting voting scenarios, that gives a unified description of several types of higher-order fuzzy sets. We further demonstrate that this framework enables us to develop fundamental measures for these higher-order fuzzy sets in an extremely streamlined and unified manner (e.g. as a result, by proving something once, you have essentially proved it for all). Thus, we believe that such a framework would be useful for the non-experts to understand and use higher-order FSTs in their application domains, and for the experts to further develop higher-order FSTs in an efficient manner.