Generalized Quadrangle

DEFINITION A generalized quadrangle is an incidence structure $(P,B,I)$ , with $I\subseteq P imes B$ an Incidence Relation , satisfying certain Axiom s. Elements of $P$ are by definition the points of the generalized quadrangle, elements of $B$ the lines. The axioms are the following: There is a $s$ ( $s\geq 1$ ) such that on every line there are exactly $s+1$ points. There is at most one point on two distinct lines. There is a $t$ ( $t\geq 1$ ) 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. DUALITY If $(P,B,I)$ is a generalized quadrangle with parameters ' $(s,t)$ ', then $(B,P,I^{-1})$ , with $I^{-1}$ the inverse incidence relation, is also a generalized quadrangle. This is the dual generalized quadrangle. Its parameters are ' $(t,s)$ '. Even if $s=t$ , the dual structure need not be isomorphic with the original structure. PROPERTIES
	<math>B	(s t+1)(t+1)</math>

	CATEGORIES ABOUT GENERALIZED QUADRANGLE

	incidence geometry

	set families

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