Jackson–Ren–Zhao semiring-variety interval problem

The Jackson–Ren–Zhao problem asks for the cardinality of specific intervals in the lattice of semiring varieties. The cited preprints claim maximal size, namely $2^{\aleph_{0}}$, for the principal intervals under discussion and attribute the original question to Jackson, Ren, and Zhao.

Known results

• A January 2025 paper reports that $\mathsf{V}(S_{7})$ has a continuum of subvarieties, with exactly six finitely based ones identified at the lattice base; it leaves broader finite-basis questions open.

January and September 2026 claimed resolution

• In January 2026, a preprint claimed that $[\mathsf{V}(S_{7}),\mathsf{V}(B_{2}^{1})]$ and the corresponding semigroup-variety interval have cardinality $2^{\aleph_{0}}$. In September 2026, Gao, Ren, and Zhao developed a general sufficient criterion, applied it to $S_{7}$ and $B_{2}^{1}$, claimed $2^{\aleph_{0}}$ intervals including $[\mathsf{V}(S_{7}),\mathsf{V}(S_{7}^{0})]$, and claimed that every variety in $[\mathsf{V}(S_{7}),\mathsf{V}((B_{2}^{1})^{0})]$ is nonfinitely based.

Current status (as of September 2026): The cited preprints claim the interval-cardinality question is solved and give related nonfinite-basis results, but these claims remain unverified.