Explicit equational bases for the power semirings of $S_7$

Abstract

For every semigroup $S$, the set $\mathcal{P}(S)$ of all subsets of $S$ and the set ${\mathcal{P}}^{+}(S)$ of all nonempty subsets of $S$ form additively idempotent semirings under set-theoretic union and elementwise multiplication, called the full and nonempty power semirings of $S$, respectively. We investigate the finite basis problem for the full and nonempty power semirings $\mathcal{P}(S_7)$ and ${\mathcal{P}}^{+}(S_7)$ of the multiplicative reduct of $S_7$, where $S_7$ is the unique nonfinitely based three-element additively idempotent semiring. We provide explicit infinite equational bases for both and prove that they are nonfinitely based. For ${\mathcal{P}}^{+}(S_7)$, we establish a new sufficient condition for an additively idempotent semiring to be nonfinitely based and apply it to obtain the required result. Moreover, we show that the interval $[\mathsf{V}({\mathcal{P}}^{+}(S_7)),\mathsf{V}(\mathcal{P}(S_7))]$ in the lattice of additively idempotent semiring varieties has the cardinality of the continuum.