The modal logic \(K_n\) which is obtained from \(K\) by replacing the usual schema \(\square \alpha \wedge \square \beta \to \square (\alpha \wedge \beta)\) with the schema of “\(n\)-ary aggregation” \[ \square \alpha_0 \wedge \cdots \wedge \square \alpha_n. \to. \square \bigwedge_{(0 \leq i < j \leq n)} (\alpha_i \wedge \alpha_j) \] is complete w.r.t. Kripke-style semantics, where in a frame \(\langle W,R \rangle\), \(R\) is an \(n\)-ary relation on \(W\), and the \(\square\)-point in the definition of truth at a world of a model is \[ \models_w \square \alpha \text{ iff } (\forall x_1, \dots, x_{n - 1} \in W) \bigl( Rw, x_1, \dots, x_{n - 1} \Rightarrow (\exists i, 1 \leq i < n) \models_{x_i} \alpha \bigr). \]