Modal meet-implication logic

We extend the meet-implication fragment of propositional intuitionistic logicwith a meet-preserving modality. We give semantics based on semilattices and aduality result with a suitable notion of descriptive frame. As a consequence weobtain completeness and identify a common (modal) fragment of a large class ofmodal intuitionistic logics. We recognise this logic as a dialgebraic logic,and as a consequence obtain expressivity-somewhere-else. Within the dialgebraicframework, we then investigate the extension of the meet-implication fragmentof propositional intuitionistic logic with a monotone modality and provecompleteness and expressivity-somewhere-else for it