Duoidal R-Matrices

In this note, we define an analogue of R-matrices for bialgebras in the setting of a monad that is opmonoidal over two tensor products. Analogous to the classical case, such structures bijectively correspond to duoidal structures on the Eilenberg–Moore category of the monad. Further, we investigate how a cocommutative version of this lifts the linearly distributive structure of a normal duoidal category.