Semiring and involution identities of power groups

TL;DR

This paper investigates the algebraic identities of power groups' semiring and involution structures, proving non-finite basis results for certain classes and solving the finite basis problem for Hall relations.

Contribution

It establishes non-finite identity bases for semirings and involution semigroups of power groups under specific conditions and solves the finite basis problem for Hall relations.

Findings

No finite identity basis for certain power groups' semiring and involution structures.

Finite basis problem solved for Hall relations over finite sets.

Results apply to finite, non-Dedekind, solvable groups.

Abstract

For every group $G$, 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 group $G$ is 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.