Motivated by better understanding the bideterminant (=product of minors) basis on the polynomial ring in $n \times m$ variables, we develop theory \& algorithms for Gr\"obner bases in not only algebras with straightening law (ASLs or Hodge algebras), but in any commutative algebra over a field that comes equipped with a notion of "monomial" (generalizing the standard monomials of ASLs) and a suitable term order. Rather than treating such an algebra $A$ as a quotient of a polynomial ring and then "lifting" ideals from $A$ to ideals in the polynomial ring, the theory we develop is entirely "native" to $A$ and its given notion of monomial. When applied to the case of bideterminants, this enables us to package several standard results on bideterminants in a clean way that enables new results. In particular, once the theory is set up, it lets us give an almost-trivial proof of a universal Gr\"obner basis (in our sense) for the ideal of $t$-minors for any $t$. We note that here it was crucial that theory be native to $A$ and its given monomial structure, as in the standard monomial structure given by bideterminants each $t$-minor is a single variable rather than a sum of $t!$ many terms (in the "ordinary monomial" structure).

翻译：为了更好地理解$n \times m$变量多项式环上的双行列式（即子式乘积）基，我们发展了一套适用于任意域上交换代数的格罗布纳基理论及算法，这些代数不仅包括具有直化律的代数（ASLs或霍奇代数），还涵盖任何配备“单项式”概念（推广ASLs的标准单项式）及合适项序的交换代数。与将此类代数$A$视为多项式环的商代数，再将理想从$A$“提升”至多项式环中的理想不同，我们所发展的理论完全“内生于”$A$及其给定的单项式概念。当应用于双行列式情形时，该理论使我们能够以清晰的方式整合关于双行列式的若干标准结果，并推导出新结果。特别地，理论建立后，它允许我们对任意$t$，给出$t$阶子式理想的一个（按我们定义的）通用格罗布纳基的近乎平凡的证明。我们指出，此处理论必须内生于$A$及其给定的单项式结构至关重要，因为在由双行列式给出的标准单项式结构中，每个$t$阶子式是单个变量，而非$t!$项之和（在“通常单项式”结构中）。