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Many-valued binary relations are considered, taking values in a complete lattice, possibly equipped with additional operations that characterize extended-order algebras. The functionality of such relations is defined, by means of properties such as univocality and totality, from three different perspectives, namely from the viewpoint of α-cuts, of composition and of powerset operators, respectively.Functional many-valued relations are generalizations of functions and special care is devoted to characterize those which are actually functions. Relationships between the involved concepts are discussed with several results and examples.