Monoidal category of operad of graphs

TL;DR

This paper develops a monoidal category framework for operads of graphs, introducing a graphical calculus and exploring applications to various logical systems including classical, modal, and many-valued logics.

Contribution

It introduces a novel monoidal category of operad of graphs with a graphical calculus and applies it to interpret various logical systems within this categorical framework.

Findings

Graphical calculus for operad of graphs developed from scratch.

Connections established between braided monoidal categories and logical algebras.

Applications demonstrated for classical, modal, and Lukasiewicz three-valued logic.

Abstract

Usually a name of the category is inherited from the name of objects. However more relevant for a category of objects and morphisms is an algebra of morphisms. Therefore we prefer to say a category of graphs if every morphism is a graph. In a monoidal category every morphism can be seen as a graph, and a partial algebra of morphisms possesses a structure of an operad, operad of graphs. We consider a monoidal category of operad of graphs with underlying graphical calculus. If, in particular, there is a single generating objects, then each morphism is a bi-arity graph. The graphical calculus, multi-grafting of morphisms, is developed ab ovo. We interpret algebraic logic and predicate calculus within a monoidal category of operad of graphs, and this leads to the graphical logic. A logic based on a braided monoidal category is said to be the braided logic. We consider a braided monoidal category generated by one object. We are demonstrating how the braided logic is related to implicative algebra and to the Heyting algebra (in contrary to the Boolean algebra) and therefore must be more related to the quasigroups then to the lattices. Some applications to classical logic, to modal logic and to Lukasiewicz three-valued logic are considered.