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Exposure Orders in Free Group Algebras: Minimal Schreier Transversals, Free Bases, and Gröbner Bases

Matan Seidel

TL;DR

The work develops a general framework of exposure orders on free groups and leverages it to construct minimal Schreier transversals, exposure bases, and Gröbner bases for any one-sided ideal in the free group algebra $ ext{A}=K[F]$. By replacing the shortlex order with computable exposure orders, it extends Lewin and Rosenmann’s results to infinitely generated ideals and provides algorithmic tools (remainder computation, division, basis construction, and canonical expression) when $I$ is finitely generated. Central notions include the minimal Schreier transversal $T_{I}$, the exposure basis $B_{I}$, the second set $ ext{s}_{eta}$, and the combinatorially reducing system (CRS) that underpins a robust Gröbner framework without requiring suffix-invariance in general. The made framework unifies prior approaches and clarifies the role of suffix-invariance, showing that algorithmic Gröbner theory in $ ext{A}$ does not fundamentally rely on it, while still enjoying simpler structure when suffix-invariant orders (e.g., shortlex) are used. The results yield practical algorithms for computing minimal structures and enable expressing elements of $I$ in terms of the exposure basis, with potential applications to problems related to word measures and primitivity testing in free groups.

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 Gröbner 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 Gröbner 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 Gröbner 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.

Exposure Orders in Free Group Algebras: Minimal Schreier Transversals, Free Bases, and Gröbner Bases

TL;DR

The work develops a general framework of exposure orders on free groups and leverages it to construct minimal Schreier transversals, exposure bases, and Gröbner bases for any one-sided ideal in the free group algebra . By replacing the shortlex order with computable exposure orders, it extends Lewin and Rosenmann’s results to infinitely generated ideals and provides algorithmic tools (remainder computation, division, basis construction, and canonical expression) when is finitely generated. Central notions include the minimal Schreier transversal , the exposure basis , the second set , and the combinatorially reducing system (CRS) that underpins a robust Gröbner framework without requiring suffix-invariance in general. The made framework unifies prior approaches and clarifies the role of suffix-invariance, showing that algorithmic Gröbner theory in does not fundamentally rely on it, while still enjoying simpler structure when suffix-invariant orders (e.g., shortlex) are used. The results yield practical algorithms for computing minimal structures and enable expressing elements of in terms of the exposure basis, with potential applications to problems related to word measures and primitivity testing in free groups.

Abstract

Consider the free group algebra , where is a free group and a field. A well-order on is called an exposure order if words are greater than their proper prefixes. We show that every one-sided ideal in admits a Schreier transversal, a basis, and a Gröbner basis -- each minimal in a natural sense with respect to . When is finitely generated and 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 and to infinitely generated ideals, while retaining algorithmic capabilities for such orders in case is finitely generated. General exposure orders lack a form of compatibility with products which we call suffix-invariance, that shortlex enjoys, and which prior Gröbner basis constructions in 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 Gröbner theory in 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 , any Schreier transversal for can be realized as minimal, and any basis for arising from the constructions of Lewin or Rosenmann can likewise be realized as its minimal basis, unifying both approaches under a single framework.

Paper Structure

This paper contains 28 sections, 36 theorems, 8 equations, 5 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Ordering Supports of $\mathcal{A}$ via Orders on $F$
  3. Minimal Schreier Transversals
  4. Minimal Bases
  5. Gröbner Bases
  6. Algorithmic Results
  7. Notation and Scope
  8. Paper Organization
  9. Preliminary Concepts
  10. Exposure orders on $F$
  11. Well-Orders and Ordinals
  12. Ordering the Subsets of $F$
  13. Induced Minimal Structures
  14. Induced Minimal Structures
  15. The Minimal Schreier Transversal of an Ideal
  16. ...and 13 more sections

Key Result

Theorem 1.2

Let $I$ be a right ideal in $\mathcal{A}$. The set of Schreier transversals for $I$ admits a minimum $T_{I,\prec}$ with respect to the order $\prec_{\text{min}}$.

Figures (5)

  • Figure 1: On the left, an illustration of Example \ref{['exa: word smaller than its phi']}: the vertex $v=xy$ has $\prec_{\text{max}}$-smaller support than its remainder $\phi_{I}\left(v\right)=y$ modulo $T_{I}$, with both vertices depicted in red. On the right, the proof of Proposition \ref{['prop: Reducing a neighbour of minimal Schreier Transversal T_I decreases support']} shows that this is impossible when $v$ is a prefix-neighbor of $T_{I}$: Suppose for contradiction that the head term $u$ in its remainder satisfies $u\succ v$. Then truncating $F_{u}$ (in purple) from $T_{I}$ and then adding $v$ (in red) results in a smaller partial Schreier transversal, contradicting $\prec_{\text{min}}$-minimality.
  • Figure 2: Illustration of Example \ref{['exa: example with no inclusion relation when increasing ideal']}: The ideal $I=\left(x-1\right)\mathcal{A}$ contains $J=\left(x-1\right)\left(y-1\right)\mathcal{A}$, but no inclusion relation exists between their respective minimal Schreier transversals $T_{I}$ (on the left, in green) and $T_{J}$ (on the right, in blue). Edges and vertices outside of each Schreier transversal are greyed out.
  • Figure 3: Schematic illustration of the support of an exposure element $f_{\alpha}$, as in Proposition \ref{['prop: Form for first using T_I']}. The support of $f_{\alpha}$ is indicated by red squares adjacent to the corresponding vertices. The final letter $b_{\alpha}$ of the head term $w_{\alpha}^{+}$ exits $T_{I}$, making $w_{\alpha}^{+}$ its prefix-neighbor. The rest of $\text{supp}\left(f_{\alpha}\right)$ lies within $T_{I}$.
  • Figure 4: Illustration for Example \ref{['examples of seconds']} (\ref{['enu: involved example of seconds']}), showing the evolution of the minimal Schreier transversal during the two-stage exposure process of the ideal $I$: Top left: the initial Schreier transversal $T_{I_{0}}$ equals the entire Cayley tree. Bottom left: following the exposure of $f_{0}=y^{2}+xy+y^{-1}$ and its corresponding second$s_{0}=y^{-2}+y+x$, the Schreier transversal $T_{I_{1}}$ excludes the prefixes $\left\{ y^{2},y^{-2}\right\}$. Bottom right: following the exposure of $f_{1}=xy^{-1}+y$ and its corresponding second$s_{1}=xy+x+y^{-1}$, the Schreier transversal $T_{I_{2}}=T_{I}$ excludes the prefixes $\left\{ y^{2},y^{-2},xy,xy^{-1}\right\}$. The supports of $f_{0},s_{0},f_{1},s_{1}$ are indicated by red squares, red triangles, green squares, and green triangles, respectively.
  • Figure 6: Illustration of Example \ref{['exa: Another option for Grobner basis']}. The combinatorially reducing system $\mathcal{Q}'=\left\{ f_{0},s_{0}',f_{1},s_{1}'\right\}$ is shown. While $s_{1}'$ coincides with the original second$s_{1}$, the element $s_{0}'$ differs from the second$s_{0}$ and no longer contains $\text{HT}\left(s_{1}'\right)=w_{1}^{-}$ in its support (cf. Figure \ref{['fig: involved example of seconds']}). The supports of $s_{0}'$ and $s_{1}'$ are indicated by red and green primed triangles, respectively.

Theorems & Definitions (83)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Claim 2.1
  • Claim 2.2
  • ...and 73 more