Semiring and involution identities of power groups. (English) Zbl 07765341
Summary: 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.
Keywords:
additively idempotent semiring; finite basis problem; power semiring; power group; block-group; Hall relation; Brandt monoid; involution semigroupReferences:
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