Partial Algebraic Theories

The goal of this talk is to arrive at a specific notion of "partial algebraic theory". Partial algebraic theories allow us to specify algebraic structures involving partial functions, much as the more classical notion of algebraic theory allows us to specify algebraic structure involving (total) functions. These partial algebraic theories are equivalent in expressive power to a number of more established approaches to specifying algebraic structure involving partial functions, including finite limit sketches, partial horn theories, and essentially algebraic theories. Unlike these more established approaches, partial algebraic theories admit a natural notion of presentation, based in string diagrams for monoidal categories.