Extensions of Steiner Triple Systems

TL;DR

This paper explores how Steiner triple systems can be extended through Steiner loops, especially Schreier extensions, to construct larger systems with Veblen points, enhancing understanding of their algebraic structure.

Contribution

It introduces a detailed study of extensions of Steiner loops, particularly Schreier extensions, to generate new Steiner triple systems with Veblen points.

Findings

Schreier extensions effectively construct larger Steiner triple systems.

Veblen points correspond to the center of Steiner loops.

Extensions reveal new structural properties of Steiner systems.

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.