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We establish a duality between two categories, extending the Stone duality between totally disconnected compact Hausdorff spaces (Stone spaces) and Boolean rings with a unit. The first category denoted by RHQS, has as objects the representations of Hansdorff quotients of Stone spaces and as morphisms all compatible continuous functions. The second category, denoted by BRLR, has as objects all Boolean rings with a unit endowed with a link relation and as morphisms all compatible Boolean rings with unit morphisms. Furthermore, we study connectedness from an algebraic point of view, in the context of the proposed generalized Stone duality.