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We introduce the notions of generalized join-hemimorphism and generalized Boolean relation as an extension of the notions of join-hemimorphism and Boolean relation, respectively. We prove a duality between these two notions. We will also define a generalization of the notion of Boolean algebra with operators by considering a finite family of Boolean algebras endowed with a generalized join-hemimorphism. Finally, we define suitable notions of subalgebra, congruences, Boolean equivalence, and open filters.