Positive monotone modal logic is the negation- and implication-free fragment of monotone modal logic, i.e., the fragment with connectives and. We axiomatise positive monotone modal logic, give monotone neighbourhood semantics based on posets, and prove soundness and completeness. The latter follows from the main result of this paper: a duality between so-called \-spaces and the algebraic semantics of positive monotone modal logic. The main technical tool is the use of coalgebra.

We investigate bisimulations for instantial neighbourhood logic and an \-indexed collection of its fragments. For each of these logics we give a Hennessy-Milner theorem and a Van Benthem-style characterisation theorem.