Functorial semantics for partial theories

Abstract

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.

String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure

“…The correspondence is well-behaved enough to extend to presentations of theories: indeed, it has been since long understood [22] that any presentation of a Lawvere theory can be seen as a presentation of a symmetric monoidal theory [13]. PROPs are more general, however, and can be used to capture partial and relational theories [18,38,10,59]. Overall, it appears that symmetric monoidal structure is the axiomatic baseline for many pertinent examples.…”

A Variety Theorem for Relational Universal Algebra

“…The idea of using string diagrams to represent terms in more general notions of algebraic theories is relatively recent, and is heavily indebted to the work of Fox [16]. The present paper can be considered a generalisation of recent work on partial theories [14] to include relations. The idea to treat cartesian bicategories of relations as theories with models in the category of sets and relations has appeared previously in [7], although no variety theorem is provided therein.…”

“…Viewed from the perspective of categorical algebra, our variety theorem characterizes definable categories [24], which are a relatively new notion without much surrounding literature. It could also be seen as contributing to the project of a unified string-diagrammatic framework for partial, relational, and classical algebraic theories aspired to in [7], and partially fulfilled by [14].…”

“…Our development is best understood in the context of recent work on partial algebraic theories [14], in which the stringdiagrammatic syntax for algebraic theories is modified to capture partial functions. This modification of the basic syntax coincides with an increase in the expressive power of the framework, corresponding roughly to the equalizer completion of a category with finite products [9].…”

“…The move to relational algebraic theories involves a further modification of the stringdiagrammatic syntax, corresponding roughly to the regular completion of a category with finite limits [10]. Put another way, in [14] the (string-diagrammatic) syntax for algebraic theories is extended to express a certain kind of equality, and the resulting terms are best understood as partial functions. In this paper, we further extend the string-diagrammatic syntax to express existential quantification, and the restuling terms are best understood as relations.…”