Weakly Aggregative Modal Logic (WAML) is a collection of disguised polyadic modal logics with -ary modalities whose arguments are all the same. WAML has some interesting applications on epistemic logic, deontic logic, and the logic of belief, and in this paper, we study some basic model theoretical aspects of WAML. Specifically, we first give a van Benthem-Rosen characterization theorem of WAML based on an intuitive notion of bisimulation. Then, in contrast to many well known normal or non-normal modal logics, we show that each basic WAML system lacks Craig interpolation. Finally, by model theoretical techniques, we show that an extension of does have Craig interpolation, as an example of amending the interpolation problem of WAML.

(Extended journal version of the LORI19 conference paper)