regular monoidal language

Definition (Monoidal regular language). A non-deterministic monoidal automaton $(\Gamma ,S,\varphi ,i,f)$ is given by a polygraph of symbols $\Gamma$, a set of states $S$, a pair of initial and final states $i,f\in S$, and a transition relation ${\varphi }_t﹕S^{\otimes n}\to S^{\otimes m}$ for each symbol $t\in \Gamma (n;m)$. Note that the transition function ${\varphi }_{n,m}﹕\Gamma (n;m)\to Rel(S^{\otimes n},S^{\otimes m})$ extends to a functor ${\varphi }^{⊛}﹕{\Gamma }^{⊛}\to Re{l}_S$.

A word $w\in {\Gamma }^{⊛}$ is accepted if there exists a transition to the final state, $f\in {\varphi }_w^{⊛}(i)$. A monoidal regular language is a set of words that arises as the set of words accepted by a monoidal automaton.

References:

• Regular Monoidal Languages (Earnshaw, Sobocinski, 2022)
• example of regular monoidal grammar
• regular language