On the completeness of the traced monoidal category axioms in (Rel,+)

Abstract

It is shown that the traced monoidal category of finite sets and relations with coproduct as tensor is complete for the extension of the traced symmetric monoidal axioms by two simple axioms, which capture the additive nature of trace in this category. The result is derived from a theorem saying that already the structure of finite partial injections as a traced monoidal category is complete for the given axioms. In practical terms this means that if two biaccessible flowchart schemes are not isomorphic, then there exists an interpretation of the schemes by partial injections which distinguishes them.