SEMIRING AND INVOLUTION IDENTITIES OF POWER GROUPS

For every group G, the set P(G) of its subsets forms a semiring under set-theoretical union u and element-wise multiplication middot, and forms an involution semigroup under middot and element-wise inversion( -1). We show that if the group G is finite, non-Dedekind, and solvable, neither the semiring (P(G), u, middot) nor the involution semigroup (P(G), middot, (-1)) admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.