Representation of fuzzy subsets by Galois connections

There is a great deal of fuzziness in our everyday natural language, and thus fuzzy subsets have come to represent a direct generalisation of the indicator function of a classical subset. On the other hand, a Galois connection is given by two opposite order-inverting maps whose composition yields two closure operations between ordered sets. We present the one-to-one correspondence between a set of all fuzzy subsets and a set of all Galois connections. The essential correspondences are built with the help of α-cuts, which represent fuzzy subsets by means of classical sets. Moreover, we present a relationship between strong fuzzy negations in the lattices and Galois connections. The various extensions of fuzzy subsets from the point of view of nestedness and negations are recalled. Other fruitful properties and connections with related studies are included.