Definition
A generalized quadrangle is an incidence structure (P,B,I), with I ⊆ P × B an incidence relation, satisfying certain axioms. Elements of P are by definition the points of the generalized quadrangle, elements of B the lines. The axioms are the following:
- There is an s (s ≥ 1) such that on every line there are exactly s + 1 points. There is at most one point on two distinct lines.
- There is a t (t ≥ 1) such that through every point there are exactly t + 1 lines. There is at most one line through two distinct points.
- For every point p not on a line L, there is a unique line M and a unique point q, such that p is on M, and q on M and L.
(s,t) are the parameters of the generalized quadrangle. The parameters are allowed to be infinite. If either s or t is one, the generalized quadrangle is called trivial. A generalized quadrangle with parameters (s,t) is often denoted by GQ(s,t).
The smallest non-trivial generalized quadrangle is GQ(2,2), whose representation has been dubbed "the doily" by Stan Payne in 1973.
Read more about this topic: Generalized Quadrangle
Famous quotes containing the word definition:
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)