Geometry

From Groupprops
Jump to: navigation, search
This term is related to: incidence geometry
View other terms related to incidence geometry | View facts related to incidence geometry

Template:Incidence system property

Definition

Definition with symbols

An incidence system (G,*,t,\Phi) is termed a geometry if it satisfies the following additional conditions:

  • Every maximal flag contains exactly one element of each type, viz every maximal flag has corank 0
  • For any two distinct types, the bipartite graph on the union of elements of those types given by the incidence relation, is a connected graph

Graph-theoretic definition

An incidence system is termed a geometry if the corresponding graph has the following properties:

  • Every maximal clique in it has one vertex from each part
  • The subgraph induced on any two parts is connected.

Relation with other properties

Weaker properties