"Spider" is a nickname of *special Frobenius algebras*, a fundamental structure from mathematics, physics, and computer science. *Pregroups* are a fundamental structure from linguistics. Pregroups and spiders have been used together in natural language processing: one for syntax, the other for semantics. It turns out that pregroups themselves can be characterized as pointed spiders in the category of preordered relations, where they naturally arise from grammars. The other way around, general spider algebras can be characterized as pregroup unions. This extends the characterization of spider algebras over sets and relations as disjoint unions of abelian groups. The compositional framework that emerged with the results suggests new ways to understand and apply the basis structures in machine learning and data analysis.

中文翻译：

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Lambek的预购者是预购中的Frobenius蜘蛛

“蜘蛛”是*特殊Frobenius代数*的昵称，它是数学，物理学和计算机科学的基础结构。*预分组*是语言学的基本结构。预分组和蜘蛛在自然语言处理中已经一起使用：一种用于语法，另一种用于语义。事实证明，在预先排序的关系类别中，预分组本身可以被描述为尖尖的蜘蛛，它们自然而然地是从语法中产生的。相反，一般的蜘蛛代数可以被描述为群前结合。这将蜘蛛代数的特征扩展到集合和关系上，作为阿贝尔群的不相交的并集。结果中出现的组成框架为理解和应用机器学习和数据分析中的基础结构提供了新的方法。