Given an element x of a commutative quasigroup $\mathcal{A}$, the x-divisor graph of $\mathcal{A}$ is the graph ${\mathrm{\Gamma }}_x(\mathcal{A})$ whose vertices are the elements of $\mathcal{A}$ such that distinct vertices a and b are adjacent if and only if $ab=x$. This paper examines how isotopies act on the set $\{{\mathrm{\Gamma }}_x(\mathcal{A})$ | $x\in \mathcal{A}\}$. It is shown that if $|\mathcal{A}|<\infty$ is not a multiple of 4, then this set is an isotopy invariant. Moreover, under certain conditions (for example, when $\mathcal{A}$ is isotopic to a group), the commutative quasigroup isotopes of $\mathcal{A}$ are completely determined by symmetries of $\{{\mathrm{\Gamma }}_x(\mathcal{A})$ | $x\in \mathcal{A}\}$.

中文翻译：

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可交换拟群的除数图和同位素不变量

给定可交换拟群的元素x$\mathcal{一个}$, x -除数图$\mathcal{一个}$ 是图 ${\mathrm{\Gamma }}_X(\mathcal{一个})$ 其顶点是 $\mathcal{一个}$使得不同的顶点a和b相邻当且仅当$\mathrm{一个}乙=X$. 本文研究了同位素如何作用于集合$\{{\mathrm{\Gamma }}_X(\mathcal{一个})$ | $X\in \mathcal{一个}\}$. 结果表明，如果$|\mathcal{一个}|<\infty$不是 4 的倍数，则该集合是同位素不变量。此外，在某些条件下（例如，当$\mathcal{一个}$ 是一个群的同位素), 的交换准群同位素 $\mathcal{一个}$ 完全由对称性决定 $\{{\mathrm{\Gamma }}_X(\mathcal{一个})$ | $X\in \mathcal{一个}\}$.