A tensor rank theory and maximum full rank subtensors

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