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.


Slice rank of a diagonal tensor

{wiki}

For finite sets $X,Y,Z$ and a field $F$, a three-variable tensor $T:X\times Y\times Z\to F$ has slice rank at most $r$ when it is a sum of $r$ terms, each of which separates one variable from the other two. Such terms have one of the forms
$$
a(x)b(y,z),\qquad c(y)d(x,z),\qquad e(z)h(x,y).
$$
The slice rank is the least possible $r$.


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