The Burnside ai-semiring variety defined by $x^n\approx x$

Let ${\bf Sr}(n, 1)$ denote the ai-semiring variety defined by the identity $x^n\approx x$, where $n>1$. We characterize all subdirectly irreducible members of a semisimple subvariety of ${\bf Sr}(n, 1)$. Based on this result, we prove that ${\bf Sr}(n, 1)$ is hereditarily finitely based (resp., hereditarily finitely generated) if and only if $n<4$ and that the lattice of subvarieties of ${\bf Sr}(n, 1)$ is countable if and only if $n<4$. Also, we show that the class of all locally finite members of ${\bf Sr}(n, 1)$ forms a variety and so we affirmatively answer the restricted Burnside problem for ${\bf Sr}(n, 1)$. In addition, we provide a simplified proof of the main result obtained by Gajdo\v{s} and Ku\v{r}il (Semigroup Forum 80: 92--104, 2010).