We study the lattice L(CSr(n,1))\documentclass[12pt]{minimal}
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\begin{document}$${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$$\end{document} of subvarieties of the ai-semiring variety CSr(n,1)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbf{CSr}}}(n, 1)$$\end{document} defined by xn≈x\documentclass[12pt]{minimal}
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\begin{document}$$x^n\approx x$$\end{document} and xy≈yx\documentclass[12pt]{minimal}
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\begin{document}$$xy\approx yx$$\end{document}. We divide L(CSr(n,1))\documentclass[12pt]{minimal}
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\begin{document}$${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$$\end{document} into five intervals and provide an explicit description of each member of these intervals except [CSr(2,1),CSr(n,1)]\documentclass[12pt]{minimal}
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\begin{document}$$[{{\mathbf{CSr}}}(2, 1), {\mathbf{CSr}}(n, 1)]$$\end{document}. Based on these results, we show that if n-1\documentclass[12pt]{minimal}
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\begin{document}$$n-1$$\end{document} is square-free, then L(CSr(n,1))\documentclass[12pt]{minimal}
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\begin{document}$${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$$\end{document} is a distributive lattice of order 2+2r+1+3r\documentclass[12pt]{minimal}
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\begin{document}$$2+2^{r+1}+3^r$$\end{document}, where r denotes the number of prime divisors of n-1\documentclass[12pt]{minimal}
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\begin{document}$$n-1$$\end{document}. Also, all members of L(CSr(n,1))\documentclass[12pt]{minimal}
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\begin{document}$${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$$\end{document} are finitely based and finitely generated and so CSr(n,1)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbf{CSr}}}(n, 1)$$\end{document} is a Cross variety. Moreover, the axiomatic rank of each member of L(CSr(n,1))\documentclass[12pt]{minimal}
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\begin{document}$${\mathscr {L}}({{\mathbf{CSr}}}(n, 1))$$\end{document} is less than four.
