Relative monads provide a controlled view of computation. We generalise the monadic metalanguage to a relative setting and give a complete semantics with strong relative monads. Adopting this perspective, we generalise two existing program calculi from the literature. We provide a linear-non-linear language for graded monads, LNL-RMM, along with a semantic proof that it is a conservative extension of the graded monadic metalanguage. Additionally, we provide a complete semantics for the arrow calculus, showing it is a restricted relative monadic metalanguage. This motivates the introduction of ARMM, a computational lambda calculus-style language for arrows that conservatively extends the arrow calculus.

@article{paperbot3939,
  title = {The Relative Monadic Metalanguage},
  author = {Jack Liell-Cock and Zev Shirazi and Sam Staton},
  journal = {Proceedings of the ACM on Programming Languages},
  volume = {10},
  number = {POPL},
  year = {2026},
  doi = {10.1145/3776702}
}