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

Let $\mathbf{S}\mathbf{r}(n,1)$ 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 $\mathbf{S}\mathbf{r}(n,1)$. Based on this result, we prove that $\mathbf{S}\mathbf{r}(n,1)$ is hereditarily finitely based (resp., hereditarily finitely generated) if and only if $n<4$ and that the lattice of subvarieties of $\mathbf{S}\mathbf{r}(n,1)$ is countable if and only if $n<4$. Also, we show that the class of all locally finite members of $\mathbf{S}\mathbf{r}(n,1)$ forms a variety and so we affirmatively answer the restricted Burnside problem for $\mathbf{S}\mathbf{r}(n,1)$. In addition, we provide a simplified proof of the main result obtained by Gajdoš and Kuřil (Semigroup Forum 80: 92–104, 2010).