Mathematical Formulae Beyond the Tree Paradigm

A restriction in the traditional language of mathematics is that every formula has an underlying tree structure, wherein an expression decomposes into subexpressions that each contribute one result to the whole. Some modern mathematics (especially in higher algebra and differential geometry) is however not so well catered for by this tree paradigm, and would be much better off if this restriction was lifted; it is not that the mathematics requires a generalised formula language, but it becomes a whole lot easier to state and do when you use one.This poster shows the formula language of networks that arises when one relaxes the tree restriction to instead allow an underlying directed acyclic graph (DAG) structure. Evaluation of networks is non-obvious, but can be carried out in any PROP/symocat; conversely, evaluation of networks characterises PROPs in much the same way as evaluation of words characterises monoids. Networks are isomorphic to the abstract index interpretation of Einstein convention tensor expressions, and demonstrate that these can in fact be given a coordinate-free interpretation. Special cases of the network notation that have been invented independently include the Penrose graphical notation and quantum gate arrays.