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