We determine the border subrank of higher order structure tensors of several families of algebras, and in particular obtain the following results. (1) We determine tight bounds on the border subrank of $k$-fold matrix multiplication and $k$-fold upper triangular matrix multiplication for all $k$. (2) We determine the border subrank of the higher order structure tensors of truncated polynomial algebras, null algebras, and apolar algebras of a quadric. (3) We determine the border subrank of the higher order structure tensors of the Lie algebra $\mathfrak{sl}_2$ for all orders. (4) We prove that degeneration of structure tensors of algebras propagates from higher to lower order. Along the way, we investigate which upper bound methods (geometric rank, $G$-stable rank, socle degree) are effective in which settings, and how they relate. Our work extends the results of Strassen (J.~Reine Angew.~Math., 1987, 1991), who determined the asymptotic subrank of these algebras for tensors of order three, in two directions: we determine the border subrank itself rather than its asymptotic version, and we consider higher order structure tensors.

翻译：我们确定了若干代数族的高阶结构张量的边界子秩，并特别获得以下结果：（1）对于所有$k$，我们给出了$k$重矩阵乘法与$k$重上三角矩阵乘法的边界子秩的紧界；（2）我们确定了截断多项式代数、零代数及二次曲面的极代数的边界子秩；（3）我们确定了所有阶数下李代数$\mathfrak{sl}_2$的高阶结构张量的边界子秩；（4）我们证明代数的结构张量的退化性会从高阶传播至低阶。在此过程中，我们探讨了哪些上界方法（几何秩、$G$-稳定秩、Socle度数）在何种设定下有效，以及它们之间的关联。本研究在以下两个方向上拓展了Strassen（J.~Reine Angew.~Math., 1987, 1991）关于三阶张量代数渐近子秩的结果：我们确定了边界子秩本身而非其渐近版本，同时考虑了高阶结构张量。