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Extensions of Steiner Triple Systems

Giovanni Falcone, Agota Figula, Mario Galici

TL;DR

The paper develops a cohesive framework to study extensions of Steiner triple systems through Steiner loops, centering on Schreier extensions and Steiner operators. It establishes when Schreier extensions yield Steiner loops via a cohomology-like theory of factor systems, and uses this to construct and classify STS containing Veblen points, linking central elements to projective geometries over $GF(2)$ and deriving order-based thresholds for projectivity. It also introduces Steiner operators as a general method to build and analyze extensions, showing that the full STS multiplication table is governed by diagonal and a small subset of off-diagonal blocks, and that equivalence/isomorphism of Steiner operators corresponds to isotopies of Latin-square blocks. Overall, the work connects Steiner loop theory, extension theory, and finite geometry to provide constructive tools for enumerating and characterizing STS with Veblen configurations, including projective cases such as $PG(n-1,2)$ and orders $19$, $27$, and $31$.

Abstract

In this article we study extensions of Steiner triple systems by means of the associated Steiner loops. We recognize that the set of Veblen points of a Steiner triple system corresponds to the center of the Steiner loop. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provide a powerful method for constructing Steiner triple systems containing Veblen points.

Extensions of Steiner Triple Systems

TL;DR

The paper develops a cohesive framework to study extensions of Steiner triple systems through Steiner loops, centering on Schreier extensions and Steiner operators. It establishes when Schreier extensions yield Steiner loops via a cohomology-like theory of factor systems, and uses this to construct and classify STS containing Veblen points, linking central elements to projective geometries over and deriving order-based thresholds for projectivity. It also introduces Steiner operators as a general method to build and analyze extensions, showing that the full STS multiplication table is governed by diagonal and a small subset of off-diagonal blocks, and that equivalence/isomorphism of Steiner operators corresponds to isotopies of Latin-square blocks. Overall, the work connects Steiner loop theory, extension theory, and finite geometry to provide constructive tools for enumerating and characterizing STS with Veblen configurations, including projective cases such as and orders , , and .

Abstract

In this article we study extensions of Steiner triple systems by means of the associated Steiner loops. We recognize that the set of Veblen points of a Steiner triple system corresponds to the center of the Steiner loop. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provide a powerful method for constructing Steiner triple systems containing Veblen points.
Paper Structure (5 sections, 18 theorems, 82 equations, 5 figures, 5 tables)

This paper contains 5 sections, 18 theorems, 82 equations, 5 figures, 5 tables.

Table of Contents

  1. General facts
  2. Steiner loops
  3. Schreier extensions of Steiner loops
  4. Equivalent and isomorphic extensions
  5. Steiner operators

Key Result

Theorem 2.2

Let $\mathcal{S}$ be a Steiner triple system and $\mathcal{L_S}$ the corresponding Steiner loop with identity $\Omega$.

Figures (5)

  • Figure 1: Pasch configuration
  • Figure 2: Fano plane generated by a normal triple and an outer point
  • Figure 3: Pasch configuration containing $a$ and $b$ but not their triple
  • Figure 4: $\mathrm{STS}(7)$$\mathcal{Q}$
  • Figure 5: $\mathrm{STS}(9)$$\mathcal{Q}$

Theorems & Definitions (53)

  • Definition 2.1
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Example 2.3.1
  • Definition 2.4
  • Example 2.4.1
  • Definition 2.5
  • Remark 2.2
  • ...and 43 more