A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix property. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery. However, their theory is still not matured yet. Can we set an axiom system for tensor ranks? Can we extend the max-full-rank-submatrix property to tensors? We explore these in this paper. We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.

中文翻译：

---

张量秩理论和最大满秩次幂

矩阵始终具有完整秩的子矩阵，以使该矩阵的秩等于该子矩阵的秩。该性质是矩阵秩理论的基石之一。我们将此属性称为max-full-rank-submatrix属性。张量秩在低秩张量逼近，张量完成和张量恢复中起着至关重要的作用。但是，他们的理论还没有成熟。我们可以为张量等级设置公理系统吗？我们可以将max-full-rank-submatrix属性扩展到张量吗？我们将在本文中进行探讨。我们首先提出张量秩函数的一些公理。然后，我们介绍适当的张量秩函数。CP等级是张量等级函数，但不合适。有两个适当的张量秩函数，max-Tucker秩和submax-Tucker秩，它们与Tucker分解相关联。我们在张量秩函数之间定义了偏序，并表明存在唯一的最小张量秩函数。我们介绍了全秩张量概念，并定义了max-full-rank-subtensor属性。我们显示max-Tucker张量秩函数和最小的张量秩函数具有此属性。我们为任意适当的张量秩函数定义了闭包，并表明它仍然是一个适当的张量秩函数并具有max-full-rank-subtensor属性。还介绍了submax-Tucker等级的应用。我们为任意适当的张量秩函数定义了闭包，并表明它仍然是一个适当的张量秩函数并具有max-full-rank-subtensor属性。还介绍了submax-Tucker等级的应用。我们为任意适当的张量秩函数定义了闭包，并表明它仍然是一个适当的张量秩函数并具有max-full-rank-subtensor属性。还介绍了submax-Tucker等级的应用。