We show how modal quantales arise as convolution algebras of functions from lr-multisemigroups that is, multisemigroups with a source map l and a target map r, into modal quantales which can be seen as weight or value algebras. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in the three algebras. The class of lr-multisemigroups introduced in this article generalises Schweizer and Sklar's function systems and object-free categories to a setting isomorphic to algebras of ternary relations as used in boolean algebras with operators and in substructural logics. Our results provide a generic construction recipe for weighted modal quantales from such multisemigroups. This is illustrated by many examples, ranging from modal algebras of weighted relations as used in fuzzy mathematics, category quantales in the tradition of category algebras or group rings, incidence algebras over partial orders, discrete and continuous weighted path algebras, weighted languages of pomsets with interfaces, and weighted languages associated with presimplicial and precubical sets. We also discuss how these results can be combined with previous ones for concurrent quantales and generalised to a setting that supports reasoning with stochastic matrices or probabilistic predicate transformers.

中文翻译：

---

lr-多半群和模态卷积代数

我们展示了模态量子如何作为函数的卷积代数从lr-多半群（即具有源映射l和目标图r的多半群）发展为模态量子，可以看作权重或价值代数。在布尔算子与算子的传统中，我们研究三个代数中代数定律之间的模态对应。本文介绍的lr-多半群的类别将Schweizer和Sklar的函数系统和无对象类别概括为三元关系代数的同构设定，如带运算符的布尔代数和子结构逻辑中所用。我们的结果为此类多半群的加权模态量化提供了通用的构造方法。许多示例都说明了这一点，从模糊数学中使用的加权关系模态代数，到 类别代数或群环的传统中的类别量子，部分阶上的入射代数，离散和连续的加权路径代数，带有接口的Pomset的加权语言以及与简单前和立方前集合相关的加权语言。我们还将讨论如何将这些结果与先前的结果合并以进行并行量化，并推广到支持使用随机矩阵或概率谓词变换器进行推理的设置。