Non-distributive positive logic as a fragment of first-order logic over semilattices

Abstract We characterize non-distributive positive logic as the fragment of a single-sorted first-order language that is preserved by new notion simulation called meet-simulation. Meet-simulations distinguish themselves from simulations because they relate pairs states one model to single another. En route this result, we use more traditional and prove Hennessy–Milner-style theorem for it, using an analogue modal saturation meet-compactness.