Endomorphisms and the Desargues Property

Abstract

It is well known that the Desargues property for a projective geometry G is equivalent with the existence of certain collineations of G. The respective collineations φ have an axis H (i.e. a hyperplane of G such that φx = x for all x ∈ H) and a center z (i.e. a point of G such that ℓ(z, x,φx) for all x ∈ G). These notions axis and center are generalized to the case where φ: G → G is any endomorphism of G and it is shown that also for this case the existence of an axis is equivalent with the existence of a center (provided that φ is non-constant).