= Slice rank of a diagonal tensor
If
$$
T(x,y,z)=\begin{cases}c_x&x=y=z,\\0&\text{otherwise},\end{cases}
$$
on 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.
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