We describe the canonical weak distributive law $\delta \colon \mathcal S \mathcal P \to \mathcal P \mathcal S$ of the powerset monad $\mathcal P$ over the $S$-left-semimodule monad $\mathcal S$, for a class of semirings $S$. We show that the composition of $\mathcal P$ with $\mathcal S$ by means of such $\delta$ yields almost the monad of convex subsets previously introduced by Jacobs: the only difference consists in the absence in Jacobs's monad of the empty convex set. We provide a handy characterisation of the canonical weak lifting of $\mathcal P$ to $\mathbb{EM}(\mathcal S)$ as well as an algebraic theory for the resulting composed monad. Finally, we restrict the composed monad to finitely generated convex subsets and we show that it is presented by an algebraic theory combining semimodules and semilattices with bottom, which are the algebras for the finite powerset monad $\mathcal P_f$.

翻译：我们描述一个半圆形 $delta\ colone S\ mathcal S\ mathcal P\ to mathcal P\ mathcal P\ mathcal P\ mathcal S$ 以美元为单位的软分配法 $delta\ colone S p\ mathcal P \ mathcal P \ mathcal P \ mathcal P \ mathcal P \ mathcal P p \ mathcal S $, 以美元为单位的半圆形 $S 。 我们用美元表示 美元为单位的 madaltad 和 美元为单位的圆形 comvex 子子 : 唯一的区别在于雅各布 的月球没有空的 comvex 套件。 我们提供了 。 美元为 $math pad modal mogarma 的软形 理论 。