  In multilinear algebra, some special classes of matrices are extended to
higher order structured tensors. The local $w$-uniqueness solution to the
linear complementarity problem can be identified by the column competent
matrix. Motivated by this $w$-uniqueness property, we introduce column
competent tensor in the context of tensor complementarity problem. We consider
some important properties. In the theory of linear complementarity problem,
column competent matrices are introduced to study local $w$-uniqueness property
of LCP solution. We present the inheritance property and invariance property of
column competent tensors. We study the tensor complementarity problem using
column competent tensors and several results are established. Some examples are
illustrated to support the results. Keywords: Tensor complementarity problem,
column competent tensor, nondegenerate tensor, $\omega$-solution. AMS subject
classifications: 90C33, 90C30, 15A69, 46G25.
