String diagrams for bimonoidal categories (a.k.a. rig categories)?

This question is answered in the affirmative in the following preprint:

• Cole Comfort, Antonin Delpeuch, Jules Hedges, Sheet diagrams for bimonoidal categories, arXiv:2010.13361

The morphisms are represented by so-called proof sheets.

Here is an example of such a proof sheet, representing the controlled-not gate using the program sheetshow which Antonin created to typeset the diagrams in the preprint:

There isn't anything like a graphical calculus really expressly dedicated to rig categories, and well-documented and proved to be coherent, but there are 'approximations' to what you are looking for:

1. Proof nets. The most well-known graphical calculus for categories with two 'operations' are

proof nets

though it seems not quite 'canonical' how they relate to string-diagrams in the most usual sense (graphical calculus for 2-categories).

2. Graphical calculi for ologs.

A recent preprint goes some way in this direction:

Evan Patterson: Knowledge Representation in Bicategories of Relations. arXiv:1706.00526

Therein one finds the following passage relevant to your questions:

(source: op. cit. p. 44; colors added)

Please note that 'distributive monoidal category' is more special than 'rig category' (the monoidal structure is assumed to by symmetric, and the symmetric monoidal structure is assumed to conincide with the category's coproduct), but it is relevant in that every instance of the former is an instance of the latter.

3. Distributive laws via monads. This is something I am currently only beginning to really understand, but I'll mention it here briefly: there is a notion of 'distributive law' formalized via

   • two specified monads on the same specified category,

   • one more specified natural transformation,

and such a situation can of course be formalized via 'usual' string diagrams; whether this is relevant to what you are looking I do not know.