Synthetic projective lines, geometric closure and AB-sets

TL;DR

This paper introduces a novel synthetic approach to abstract projective lines, exploring geometric closures, automorphism-blocking sets, and providing examples, thereby advancing the understanding of projective geometry in an abstract setting.

Contribution

It presents a new synthetic framework for projective lines, compares it with classical methods, and introduces automorphism-blocking sets and various examples, including infinite cases.

Findings

New synthetic approach to projective lines

Introduction of automorphism-blocking sets

Construction of examples and counterexamples in infinite cases

Abstract

In this note, we introduce a new approach to abstract ``synthetic'' projective lines. We discuss various aspects of our approach, and compare these aspects with the classical one. A number of intriguing questions arise. Amongst these aspects, we discuss geometric closures, and introduce automorphism-blocking sets. We also construct a number of (counter) examples in infinite cases.