We study the canonical weak distributive law $\delta$ of the powerset monad over the semimodule monad for a certain class of semirings containing, in particular, positive semifields. For this subclass we characterise $\delta$ as a convex closure in the free semimodule of a set. Using the abstract theory of weak distributive laws, we compose the powerset and the semimodule monads via $\delta$, obtaining the monad of convex subsets of the free semimodule.

@article{paperbot1641,
  title = {Convexity via Weak Distributive Laws},
  author = {Filippo Bonchi and Alessio Santamaria},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 18, Issue 4},
  year = {2022},
  doi = {10.46298/lmcs-18(4:8)2022}
}