Modal logics with the intersection modality

We give a simple proof of a recently obtained in [12] result on the completeness of modal logics with the modality that corresponds to the intersection of accessibility relations in a Kripke model. In epistemic logic, this is the so-called distributed knowledge operator. We prove completeness for the logics in the modal languages of two types: one has the modalities $\square_1,\dots,\square_n$ for the relations $R_1,\dots,R_n$ that satisfy a unimodal logic $L$, and the modality $\square_{n+1}$ for the intersection $R_{n+1}=R_1\cap\dots\cap R_n$; the other language has the modalities $\square_i$, $(i\in\Sigma)$ for the relations $R_j$ that satisfy the logic $L_j$, and, for every nonempty subset of indices $I\subseteq\Sigma$, the modality $\square_j$ for the intersection $\bigcap_{i\in I} R_i$. While in [12] the completeness is proved only for the logics over $K$, $KD$, $KT$, $KB$, $S4$ and $S5$, here we give a “uniform” construction that enables us to obtain completeness for the logics with intersection over the 15 so-called “traditional” modal logics $K\wedge$, for $\wedge\subseteq\{D, T, B, 4, 5\}$. The proof method is based on unravelling of a frame and then taking the Horn closure of the resulting frame.