SEMIRING AND INVOLUTION IDENTITIES OF POWER GROUPS

Abstract

AbstractFor every groupG, the set$\mathcal {P}(G)$of its subsets forms a semiring under set-theoretical union$\cup $and element-wise multiplication$\cdot $, and forms an involution semigroup under$\cdot $and element-wise inversion${}^{-1}$. We show that if the groupGis finite, non-Dedekind, and solvable, neither the semiring$(\mathcal {P}(G),\cup ,\cdot )$nor the involution semigroup$(\mathcal {P}(G),\cdot ,{}^{-1})$admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set.