论文标题
关于$（$ \ Mathbb {Z}/ M \ Mathbb {Z} $，\ CDOT）$的功率序列引起的结构
On structures induced by the power sequences of $($\mathbb{Z}/ m\mathbb{Z}$, \cdot)$
论文作者
论文摘要
在本文中，我们探讨了$ \ mathbb {z}/ m \ mathbb {z} $在模块化指数下的轨道方面的结构，并用一个顺序的功率图来说明这一点，该序列图是通过连接$ \ mathbb {z}/ m \ mathbb {z} $的$ \ mathbb {z}/ m mathbb {z}/ mathbb {z} $的顺序衍生的。 我们发现该图具有大量引人入胜的代数结构。连接的组件由所有共享至少一个元素的轨道组成。连接组件的顶点集显示取决于$ m $的分解；实际上，连接的组件完全由$ \ mathbb {z}/ m \ mathbb {z} $的单位确定，$ \ mathbb {z}/ m \ mathbb {z} $和$ m $ $ m $的无方形划分。组件的尾巴和非尾部都可以根据$ \ mathbb {z}/ m \ mathbb {z} $的这些元素来明确和代数描述。最后，可以使用组件的晶格来显示晶格中任意两个可比组件的非尾部之间的同态。 在这里，这种广泛的结构被用来证明在iDempotent mod $ m $的根上具有代数身份，并且也可以利用以证明其他身份。
In this paper, we explore the structure of $\mathbb{Z}/ m\mathbb{Z}$ in terms of its orbits under modular exponentiation, illustrating this with a sequential power graph that is naturally derived from the orbits by connecting elements of $\mathbb{Z}/ m\mathbb{Z}$ in the orbit order in which they appear. We find that this graph has a great deal of fascinating algebraic structure. The connected components are composed of orbits that all share at least one element. The vertex sets of the connected components are shown to depend on the factorization of $m$; in fact, the connected components are completely determined by the units of $\mathbb{Z}/ m\mathbb{Z}$, the idempotents of $\mathbb{Z}/ m\mathbb{Z}$ and the square-free divisors of $m$. Both tails and non-tails of the components can be described explicitly and algebraically in terms of these elements of $\mathbb{Z}/ m\mathbb{Z}$. Finally, a lattice of components can be used to show homomorphisms between the non-tails of any two comparable components in the lattice. This extensive structure is used here to prove an algebraic identity on the roots of an idempotent mod $m$, and may be exploited to prove other identities as well.
