We provide a Lawvere-style definition for partial theories, extending the classical notion of equational theory by allowing partially defined operations. As in the classical case, our definition is syntactic: we use an appropriate class of string diagrams as terms. This allows for equational reasoning about the class of models defined by a partial theory. We demonstrate the expressivity of such equational theories by considering a number of examples, including partial combinatory algebras and cartesian closed categories. Moreover, despite the increase in expressivity of the syntax we retain a well-behaved notion of semantics: we show that our categories of models are precisely locally finitely presentable categories, and that free models exist.

中文翻译：

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偏理论的函数语义

我们为部分理论提供了 Lawvere 风格的定义，通过允许部分定义的操作扩展了方程理论的经典概念。与经典案例一样，我们的定义是语法上的：我们使用适当的字符串图类作为术语。这允许对由偏理论定义的模型类进行等式推理。我们通过考虑一些例子来证明这种方程理论的表现力，包括部分组合代数和笛卡尔封闭范畴。此外，尽管语法的表达能力有所提高，但我们保留了一个良好的语义概念：我们表明我们的模型类别是精确局部有限可呈现的类别，并且存在自由模型。