On the Slice Rank of Tensors in P-Echelon Form

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 $T:A^d⟶\mathbb{F}$ 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.