In this paper, we define algebraic semantics for interpretability logics, which are a family of logics which extend modal provability logic \(\textbf{GL}\) and which are aimed to formalize the notion of relative interpretability between arithmetical theories. The standard Kripke-like semantics for these logics, called Veltman semantics, lacks completeness for some extensions of the basic system \(\textbf{IL}\) . We define the notion of interpretability algebras and we show that Veltman semantics is just a special case of this semantics by showing that each Veltman frame corresponds to a particular interpretability algebra. We also show that the basic system \(\textbf{IL}\) is complete with respect to the class of all interpretability algebras defined in this paper. Moreover, every extension of \(\textbf{IL}\) is sound and complete with respect to an appropriate class of interpretability algebras.
