On the Slice Rank of Tensors in P-Echelon Form

Abstract

For a totally ordered finite set $A$ with the total order $Q$, a poset $P=([d],\le_P)$, and a field $\mathbb{F}$, a tensor $T:A^d\longrightarrow \mathbb{F}$ is in $P$-echelon form if $r\le_{P}s$ in $P$ implies $a_r\le_Q a_s$ in every tuple $(a_1,\dots,a_d)$ in the support of $T$. We prove that if the Hasse diagram of $P$ has no isolated vertex, then tensors in $P$-echelon form with nonzero diagonal entries have full slice-rank. Our results extend and improve on recent results of Amanov and Yeliussizov.