This language of matching brackets is also called the Dyck language and its syntactic monoid is the bicyclic semigroup, which is an interesting example of an inverse semigroup. Edward Kmett gave a talk on programming with these sorts of algebraic structures that some might find illuminating: https://www.youtube.com/watch?v=HGi5AxmQUwU
Inverse semigroups are used in the geometry of interaction to give a parallelizable interpretation of lambda calculus. I don't have a great introductory reference. Here's some people who have implemented a system based on the theory: https://www.cambridge.org/core/journals/mathematical-structu...
Sorry for the citation dump, I'm short on time and I thought they would be interesting to those who found this submission interesting.
Inverse semigroups are used in the geometry of interaction to give a parallelizable interpretation of lambda calculus. I don't have a great introductory reference. Here's some people who have implemented a system based on the theory: https://www.cambridge.org/core/journals/mathematical-structu...
Sorry for the citation dump, I'm short on time and I thought they would be interesting to those who found this submission interesting.
References
https://en.wikipedia.org/wiki/Dyck_language
https://en.wikipedia.org/wiki/Bicyclic_semigroup