  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.
