Diagonal tensor

A three-variable tensor is diagonal when it vanishes unless its three arguments are equal.

Ifon a finite set $X$, and every $c_x$ is nonzero, then the slice rank of $T$ is $|X|$. The upper bound uses one $x$-slice for each diagonal entry. For the lower bound, restrict any shorter slice decomposition to a common kernel of the coefficient functions in two slice directions; the remaining diagonal matrix has rank larger than the number of slices available in the third direction, a contradiction.