In residuated binars there are six non-obvious distributivity identities of ⋅,/,∖ over ∧,∨. We show that in residuated binars with distributive lattice reducts there are some dependencies among these identities; specifically, there are six pairs of identities that imply another one of these identities, and we provide counterexamples to show that no other dependencies exist among these.
Fussner, W., Jipsen, P. Distributive laws in residuated binars. Algebra Univers. 80, 54 (2019). https://doi.org/10.1007/s00012-019-0625-1