  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.
