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

中文翻译：

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满足 $$x^n\approx x$$xn≈x 和 $$xy\approx yx$$xy≈yx 的 ai-semiring 变体格

我们研究了 ai-semiring 变种 $${{\mathbf{CSr} 的子变种的格 $${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$$ }}(n, 1)$$ 由 $$x^n\approx x$$ 和 $$xy\approx yx$$ 定义。我们将 $${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$$ 划分为五个区间，并提供这些区间的每个成员的明确描述，除了 $$[{ {\mathbf{CSr}}}(2, 1), {\mathbf{CSr}}(n, 1)]$$ 。基于这些结果，我们证明如果 $$n-1$$ 是无平方的，那么 $${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$ $ 是 $$2+2^{r+1}+3^r$$ 阶的分配格，其中 r 表示 $$n-1$$ 的素因数的个数。此外，$${{\mathscr {L}}}({{\mathbf{CSr}}}(n, 1))$$ 的所有成员都是有限基和有限生成的，因此 $${{\mathbf{CSr }}}(n, 1)$$ 是一个交叉变体。而且，