Factor Geometries and Congruence Relations

Abstract

In this chapter the family of halfspaces of a nonempty linear space is converted into a join system—called a factor geometry—by defining a join operation in it in a natural way. The theory of factor geometries has its roots in the problem of constructing a geometry out of the family of rays that emanate from a given point in a Euclidean space. Such a ray geometry is implicit in classical geometry and is closely related to spherical geometry. Factor geometries and join geometries share many common properties and can be studied by similar methods. Thus in a factor geometry convexity and linearity are treated in a familiar way. However a factor geometry, as an algebraic system, differs markedly from a join geometry since it contains an identity element and its elements have inverses. The development has strong—though unforced—analogies with algebraic theories of congruence relations and factor or quotient systems.

The prerequisites for this chapter are Chapters 6, 8, Sections 9.1–9.4 and Sections 9.13–9.17.