  In the study of algebras related to non-classical logics, (distributive)
semilattices are always present in the background. For example, the algebraic
semantic of the $\{\rightarrow,\wedge,\top\}$-fragment of intuitionistic logic
is the variety of implicative meet-semilattices \cite{CelaniImplicative}
\cite{ChajdaHalasKuhr}. In this paper we introduce and study the class of
distributive meet-semilattices endowed with a monotonic modal operator $m$. We
study the representation theory of these algebras using the theory of canonical
extensions and we give a topological duality for them. Also, we show how our
new duality extends to some particular subclasses.
