论文标题
结合半模块
Combining Semilattices and Semimodules
论文作者
论文摘要
我们描述了PowerSet Monad $ \ Mathcal $ \ Mathcal p $ Mathcal p \ Mathcal P \ Mathcal p \ Mathcal p \ Mathcal $ \ Mathcal p $ p $ $ s $ s $ left-semimodule monad $ \ mathcal s $，用于$ s $ $ s $。我们表明，通过这种$δ$，$ \ Mathcal P $与$ \ Mathcal s $的组成几乎产生了雅各布斯先前引入的凸子集的单元：唯一的差异是在雅各布斯在空孔库中的雅各布斯单元中缺席。我们提供了$ \ Mathcal p $ to $ \ mathbb {em}（\ Mathcal s）$的规范弱提升的方便表征，以及用于由此产生的单调的代数理论。最后，我们将组合的单元限制为有限生成的凸子集，并表明它是由将半模块和半层次与底部结合的代数理论提出的，这是有限的Powerset Monad $ \ Mathcal p_f $的代数。
We describe the canonical weak distributive law $δ\colon \mathcal S \mathcal P \to \mathcal P \mathcal S$ 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 $δ$ 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 $\mathcal P_f$.
