The equality $(\mc{A}\n\mc{B})^\dagger = \mc{B}^\dagger \n \mc{A}^\dagger$ for any two complex tensors $\mc{A}$ and $\mc{B}$ of arbitrary order, is called as the {\it reverse-order law} for the Moore-Penrose inverse of arbitrary order tensors via the Einstein product. Panigrahy {\it et al.} [Linear Multilinear Algebra; 68 (2020), 246-264.] obtained several necessary and sufficient conditions to hold the reverse-order law for the Moore-Penrose inverse of even-order tensors via the Einstein product, very recently. This notion is revisited here among other results. In this context, we present several new characterizations of the reverse-order law of arbitrary order tensors via the same product. More importantly, we illustrate a result on the Moore-Penrose inverse of a sum of two tensors as an application of the reverse-order law which leaves an open problem. We recall the definition of the Frobenius norm and the spectral norm to illustrate a result for finding the additive perturbation bounds of the Moore-Penrose inverse under the Frobenius norm. We conclude our paper with the introduction of the notion of sub-proper splitting for tensors which may help to find an iterative solution of a tensor multilinear system.

中文翻译：

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张量的逆序定律及其在摩尔-彭罗斯逆上的加法结果中的应用

任何两个复张量 $\mc{A 的等式 $(\mc{A}\n\mc{B})^\dagger = \mc{B}^\dagger \n \mc{A}^\dagger$ }$ 和 $\mc{B}$ 的任意阶数，通过爱因斯坦乘积被称为任意阶张量的 Moore-Penrose 逆的 {\it 逆序定律}。Panigrahy {\it et al.} [线性多线性代数；68 (2020), 246-264.] 最近通过爱因斯坦乘积获得了几个充分必要条件来保持偶数阶张量的摩尔-彭罗斯逆的逆序定律。在其他结果中，这里重新讨论了这个概念。在这种情况下，我们通过相同的乘积展示了任意阶张量的逆序定律的几个新特征。更重要的是，我们将说明两个张量之和的 Moore-Penrose 逆的结果，作为逆序定律的应用，这留下了一个悬而未决的问题。我们回顾 Frobenius 范数和谱范数的定义，以说明在 Frobenius 范数下找到 Moore-Penrose 逆的加性扰动界的结果。我们最后介绍了张量的次适当分裂的概念，这可能有助于找到张量多线性系统的迭代解。