Exposure Orders in Free Group Algebras: Minimal Schreier Transversals, Free Bases, and Gr\"obner Bases

Abstract

Consider the free group algebra $K\left[F\right]$, where $F$ is a free group and $K$ a field. A well-order $\prec$ on $F$ is called an exposure order if words are greater than their proper prefixes. We show that every one-sided ideal $I$ in $K\left[F\right]$ admits a Schreier transversal, a basis, and a Gr\"obner basis -- each minimal in a natural sense with respect to $\prec$. When $I$ is finitely generated and $\prec$ is computable, we provide an algorithm for computing these minimal structures from a finite generating set. This extends the foundational works of Lewin and Rosenmann, which relied on the shortlex order, to both a broader class of orders on $F$ and to infinitely generated ideals, while retaining algorithmic capabilities for such orders in case $I$ is finitely generated. General exposure orders lack a form of compatibility with products which we call suffix-invariance, that shortlex enjoys, and which prior Gr\"obner basis constructions in $K\left[F\right]$ relied on. In its absence, reductions may strictly increase the support of elements, requiring nontrivial conceptual adaptations to definitions and algorithms. These adaptations clarify the notion of minimality underlying prior constructions and demonstrate that algorithmic Gr\"obner theory in $K\left[F\right]$ does not fundamentally require suffix-invariance, although its presence -- as in shortlex -- results in a simpler theory. Our framework further illuminates the flexibility of exposure orders: with a suitable choice of $\prec$, any Schreier transversal for $I$ can be realized as minimal, and any basis for $I$ arising from the constructions of Lewin or Rosenmann can likewise be realized as its minimal basis, unifying both approaches under a single framework.