On the logic of distributive nearlattices

We study the propositional logic $S_{\mathrm{DN}}$ associated with the variety of distributive nearlattices $\mathrm{DN}$. We prove that the logic $S_{\mathrm{DN}}$ coincides with the assertional logic associated with the variety $\mathrm{DN}$ and with the order-based logic associated with $\mathrm{DN}$. We obtain a characterization of the reduced matrix models of logic $S_{\mathrm{DN}}$. We develop a connection between the logic $S_{\mathrm{DN}}$ and the $\{\wedge ,\vee ,\top \}$-fragment of classical logic. Finally, we present two Hilbert-style axiomatizations for the logic $S_{\mathrm{DN}}$.