Quotients of incidence geometries

TL;DR

This paper develops a comprehensive theory for quotients of incidence geometries, establishing conditions for when quotients preserve geometric properties and exploring their structural implications.

Contribution

It introduces new conditions for quotients to remain geometries and analyzes their effects on properties like connectivity and transitivity, extending prior work.

Findings

Conditions for quotient geometries established

Coset pregeometries are closed under quotienting

Preservation of connectivity and transitivity under quotients

Abstract

We develop a theory for quotients of geometries and obtain sufficient conditions for the quotient of a geometry to be a geometry. These conditions are compared with earlier work on quotients, in particular by Pasini and Tits. We also explore geometric properties such as connectivity, firmness and transitivity conditions to determine when they are preserved under the quotienting operation. We show that the class of coset pregeometries, which contains all flag-transitive geometries, is closed under an appropriate quotienting operation.