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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 calledas the {\it reverse-order law} for the Moore-Penrose inverse of arbitrary ordertensors via the Einstein product. Panigrahy {\it et al.} [Linear MultilinearAlgebra; 68 (2020), 246-264.] obtained several necessary and sufficientconditions to hold the reverse-order law for the Moore-Penrose inverse ofeven-order tensors via the Einstein product, very recently. This notion isrevisited here among other results. In this context, we present several newcharacterizations of the reverse-order law of arbitrary order tensors via thesame product. More importantly, we illustrate a result on the Moore-Penroseinverse of a sum of two tensors as an application of the reverse-order lawwhich leaves an open problem. We recall the definition of the Frobenius normand the spectral norm to illustrate a result for finding the additiveperturbation bounds of the Moore-Penrose inverse under the Frobenius norm. Weconclude our paper with the introduction of the notion of sub-proper splittingfor tensors which may help to find an iterative solution of a tensormultilinear system.

On reverse-order law of tensors and its application to additive results on Moore-Penrose inverse