Hardness of learning loops, monoids, and semirings

We show that each randomized o(|G|2)-query algorithm can recover only an expected o(1) fraction of the Cayley table of some finite Abelian loop (G,⋅), where both multiplication and inversion queries are allowed. Furthermore, each randomized o(|R|2)-query algorithm can recover only an expected o(1) fraction of any of the Cayley tables of some finite commutative semiring (R,+,⋅), with (R,+) being a commutative aperiodic monoid, where each query may ask for x+y or x⋅y for any x, y∈R.