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In this paper, we explore the structure of $\mathbb{Z}/ m\mathbb{Z}$ in termsof its orbits under modular exponentiation, illustrating this with a sequentialpower 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 oneelement. The vertex sets of the connected components are shown to depend on thefactorization of $m$; in fact, the connected components are completelydetermined 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 andnon-tails of the components can be described explicitly and algebraically interms of these elements of $\mathbb{Z}/ m\mathbb{Z}$. Finally, a lattice ofcomponents can be used to show homomorphisms between the non-tails of any twocomparable components in the lattice.  This extensive structure is used here to prove an algebraic identity on theroots of an idempotent mod $m$, and may be exploited to prove other identitiesas well.

On structures induced by the power sequences of $($\mathbb{Z}/ m\mathbb{Z}$, \cdot)$