Semifinite Generalized Quadrangles G. Eric Moorhouse Department of - PowerPoint PPT Presentation

Semifinite Generalized Quadrangles G. Eric Moorhouse Department of Mathematics University of Wyoming RMAC Seminar—10 October 2014 G. Eric Moorhouse Semifinite Generalized Quadrangles

Generalized Quadrangles A generalized quadrangle (GQ) is a point-line incidence structure in which every non-incident point-line pair ( P , ℓ ) has exactly one line through P meeting ℓ : We assume every point is on more than two lines; and every line has more than 2 points. G. Eric Moorhouse Semifinite Generalized Quadrangles

Semifinite Generalized Quadrangles We say the GQ is semifinite if it has infinitely many points and lines, but the number of points on each line (always the same number) is k < ∞ . ( Open question: Can this happen? ) There is no semifinite GQ with line size k = 3 (Cameron, 1981 . . . one paragraph). There is no semifinite GQ with line size k = 4 (Brouwer, 1991 . . . three pages). There is no semifinite GQ with line size k = 5 (Cherlin, 2005 . . . seven pages of model theory). Nothing is known for line size k � 6. Experts differ on whether semifinite GQ’s may exist at all. G. Eric Moorhouse Semifinite Generalized Quadrangles

Semifinite Generalized Quadrangles We say the GQ is semifinite if it has infinitely many points and lines, but the number of points on each line (always the same number) is k < ∞ . ( Open question: Can this happen? ) There is no semifinite GQ with line size k = 3 (Cameron, 1981 . . . one paragraph). There is no semifinite GQ with line size k = 4 (Brouwer, 1991 . . . three pages). There is no semifinite GQ with line size k = 5 (Cherlin, 2005 . . . seven pages of model theory). Nothing is known for line size k � 6. Experts differ on whether semifinite GQ’s may exist at all. G. Eric Moorhouse Semifinite Generalized Quadrangles

Semifinite Generalized Quadrangles We say the GQ is semifinite if it has infinitely many points and lines, but the number of points on each line (always the same number) is k < ∞ . ( Open question: Can this happen? ) There is no semifinite GQ with line size k = 3 (Cameron, 1981 . . . one paragraph). There is no semifinite GQ with line size k = 4 (Brouwer, 1991 . . . three pages). There is no semifinite GQ with line size k = 5 (Cherlin, 2005 . . . seven pages of model theory). Nothing is known for line size k � 6. Experts differ on whether semifinite GQ’s may exist at all. G. Eric Moorhouse Semifinite Generalized Quadrangles

Semifinite Generalized Quadrangles We say the GQ is semifinite if it has infinitely many points and lines, but the number of points on each line (always the same number) is k < ∞ . ( Open question: Can this happen? ) There is no semifinite GQ with line size k = 3 (Cameron, 1981 . . . one paragraph). There is no semifinite GQ with line size k = 4 (Brouwer, 1991 . . . three pages). There is no semifinite GQ with line size k = 5 (Cherlin, 2005 . . . seven pages of model theory). Nothing is known for line size k � 6. Experts differ on whether semifinite GQ’s may exist at all. G. Eric Moorhouse Semifinite Generalized Quadrangles

Semifinite Generalized Quadrangles We say the GQ is semifinite if it has infinitely many points and lines, but the number of points on each line (always the same number) is k < ∞ . ( Open question: Can this happen? ) There is no semifinite GQ with line size k = 3 (Cameron, 1981 . . . one paragraph). There is no semifinite GQ with line size k = 4 (Brouwer, 1991 . . . three pages). There is no semifinite GQ with line size k = 5 (Cherlin, 2005 . . . seven pages of model theory). Nothing is known for line size k � 6. Experts differ on whether semifinite GQ’s may exist at all. G. Eric Moorhouse Semifinite Generalized Quadrangles

First-Order Axioms for Generalized Quadrangles P ( x ) , L ( x ) : unary predicates I ( x , y ) : binary predicate ( ∀ x )( P ( x ) ↔ ¬ L ( x )) 1 � � ( ∀ x )( ∀ y ) I ( x , y ) → ( P ( x ) ∧ L ( y )) 2 � � ( ∀ p )( ∀ ℓ ) ( P ( p ) ∧ L ( ℓ )) → ( ∃ q )( ∃ m )( I ( p , m ) ∧ I ( q , m ) ∧ I ( q , ℓ ) 3 ( ∀ p 1 )( ∀ p 2 )( ∀ ℓ 1 )( ∀ ℓ 2 ) 4 � � ( I ( p 1 , ℓ 1 ) ∧ I ( p 1 , ℓ 2 ) ∧ I ( p 2 , ℓ 1 ) ∧ I ( p 2 , ℓ 2 )) → ( p 1 = p 2 ∧ ℓ 1 = ℓ 2 ) � ( ∀ p 1 )( ∀ p 2 )( ∀ p 3 )( ∀ ℓ 1 )( ∀ ℓ 2 )( ∀ ℓ 3 ) ( I ( p 1 , ℓ 1 ) ∧ I ( p 1 , ℓ 2 ) ∧ 5 I ( p 2 , ℓ 2 ) ∧ I ( p 2 , ℓ 3 ) ∧ I ( p 3 , ℓ 3 ) ∧ I ( p 3 , ℓ 1 )) → � ( p 1 = p 2 ∧ p 1 = p 3 ∧ p 2 = p 3 ∧ ℓ 1 = ℓ 2 ∧ ℓ 1 = ℓ 3 ∧ ℓ 2 = ℓ 3 ) ( ∃ p 1 )( ∃ p 2 )( ∃ p 3 )( ∃ ℓ ) 6 � ( I ( p 1 , ℓ ) ∧ I ( p 2 , ℓ ) ∧ I ( p 3 , ℓ ) ∧ p 1 � = p 2 ∧ p 1 � = p 3 ∧ p 2 � = p 3 similarly, each point is on at least 3 lines 7 G. Eric Moorhouse Semifinite Generalized Quadrangles

First-Order Axioms for Generalized Quadrangles P ( x ) , L ( x ) : unary predicates I ( x , y ) : binary predicate ( ∀ x )( P ( x ) ↔ ¬ L ( x )) 1 � � ( ∀ x )( ∀ y ) I ( x , y ) → ( P ( x ) ∧ L ( y )) 2 � � ( ∀ p )( ∀ ℓ ) ( P ( p ) ∧ L ( ℓ )) → ( ∃ q )( ∃ m )( I ( p , m ) ∧ I ( q , m ) ∧ I ( q , ℓ ) 3 ( ∀ p 1 )( ∀ p 2 )( ∀ ℓ 1 )( ∀ ℓ 2 ) 4 � � ( I ( p 1 , ℓ 1 ) ∧ I ( p 1 , ℓ 2 ) ∧ I ( p 2 , ℓ 1 ) ∧ I ( p 2 , ℓ 2 )) → ( p 1 = p 2 ∧ ℓ 1 = ℓ 2 ) � ( ∀ p 1 )( ∀ p 2 )( ∀ p 3 )( ∀ ℓ 1 )( ∀ ℓ 2 )( ∀ ℓ 3 ) ( I ( p 1 , ℓ 1 ) ∧ I ( p 1 , ℓ 2 ) ∧ 5 I ( p 2 , ℓ 2 ) ∧ I ( p 2 , ℓ 3 ) ∧ I ( p 3 , ℓ 3 ) ∧ I ( p 3 , ℓ 1 )) → � ( p 1 = p 2 ∧ p 1 = p 3 ∧ p 2 = p 3 ∧ ℓ 1 = ℓ 2 ∧ ℓ 1 = ℓ 3 ∧ ℓ 2 = ℓ 3 ) ( ∃ p 1 )( ∃ p 2 )( ∃ p 3 )( ∃ ℓ ) 6 � ( I ( p 1 , ℓ ) ∧ I ( p 2 , ℓ ) ∧ I ( p 3 , ℓ ) ∧ p 1 � = p 2 ∧ p 1 � = p 3 ∧ p 2 � = p 3 similarly, each point is on at least 3 lines 7 G. Eric Moorhouse Semifinite Generalized Quadrangles

First-Order Axioms for Generalized Quadrangles P ( x ) , L ( x ) : unary predicates I ( x , y ) : binary predicate ( ∀ x )( P ( x ) ↔ ¬ L ( x )) 1 � � ( ∀ x )( ∀ y ) I ( x , y ) → ( P ( x ) ∧ L ( y )) 2 � � ( ∀ p )( ∀ ℓ ) ( P ( p ) ∧ L ( ℓ )) → ( ∃ q )( ∃ m )( I ( p , m ) ∧ I ( q , m ) ∧ I ( q , ℓ ) 3 ( ∀ p 1 )( ∀ p 2 )( ∀ ℓ 1 )( ∀ ℓ 2 ) 4 � � ( I ( p 1 , ℓ 1 ) ∧ I ( p 1 , ℓ 2 ) ∧ I ( p 2 , ℓ 1 ) ∧ I ( p 2 , ℓ 2 )) → ( p 1 = p 2 ∧ ℓ 1 = ℓ 2 ) � ( ∀ p 1 )( ∀ p 2 )( ∀ p 3 )( ∀ ℓ 1 )( ∀ ℓ 2 )( ∀ ℓ 3 ) ( I ( p 1 , ℓ 1 ) ∧ I ( p 1 , ℓ 2 ) ∧ 5 I ( p 2 , ℓ 2 ) ∧ I ( p 2 , ℓ 3 ) ∧ I ( p 3 , ℓ 3 ) ∧ I ( p 3 , ℓ 1 )) → � ( p 1 = p 2 ∧ p 1 = p 3 ∧ p 2 = p 3 ∧ ℓ 1 = ℓ 2 ∧ ℓ 1 = ℓ 3 ∧ ℓ 2 = ℓ 3 ) ( ∃ p 1 )( ∃ p 2 )( ∃ p 3 )( ∃ ℓ ) 6 � ( I ( p 1 , ℓ ) ∧ I ( p 2 , ℓ ) ∧ I ( p 3 , ℓ ) ∧ p 1 � = p 2 ∧ p 1 � = p 3 ∧ p 2 � = p 3 similarly, each point is on at least 3 lines 7 G. Eric Moorhouse Semifinite Generalized Quadrangles

First-Order Axioms for Generalized Quadrangles P ( x ) , L ( x ) : unary predicates I ( x , y ) : binary predicate ( ∀ x )( P ( x ) ↔ ¬ L ( x )) 1 � � ( ∀ x )( ∀ y ) I ( x , y ) → ( P ( x ) ∧ L ( y )) 2 � � ( ∀ p )( ∀ ℓ ) ( P ( p ) ∧ L ( ℓ )) → ( ∃ q )( ∃ m )( I ( p , m ) ∧ I ( q , m ) ∧ I ( q , ℓ ) 3 ( ∀ p 1 )( ∀ p 2 )( ∀ ℓ 1 )( ∀ ℓ 2 ) 4 � � ( I ( p 1 , ℓ 1 ) ∧ I ( p 1 , ℓ 2 ) ∧ I ( p 2 , ℓ 1 ) ∧ I ( p 2 , ℓ 2 )) → ( p 1 = p 2 ∧ ℓ 1 = ℓ 2 ) � ( ∀ p 1 )( ∀ p 2 )( ∀ p 3 )( ∀ ℓ 1 )( ∀ ℓ 2 )( ∀ ℓ 3 ) ( I ( p 1 , ℓ 1 ) ∧ I ( p 1 , ℓ 2 ) ∧ 5 I ( p 2 , ℓ 2 ) ∧ I ( p 2 , ℓ 3 ) ∧ I ( p 3 , ℓ 3 ) ∧ I ( p 3 , ℓ 1 )) → � ( p 1 = p 2 ∧ p 1 = p 3 ∧ p 2 = p 3 ∧ ℓ 1 = ℓ 2 ∧ ℓ 1 = ℓ 3 ∧ ℓ 2 = ℓ 3 ) ( ∃ p 1 )( ∃ p 2 )( ∃ p 3 )( ∃ ℓ ) 6 � ( I ( p 1 , ℓ ) ∧ I ( p 2 , ℓ ) ∧ I ( p 3 , ℓ ) ∧ p 1 � = p 2 ∧ p 1 � = p 3 ∧ p 2 � = p 3 similarly, each point is on at least 3 lines 7 G. Eric Moorhouse Semifinite Generalized Quadrangles

First-Order Axioms for Generalized Quadrangles To say that the line size is 3, add an axiom • ( ∀ p 1 )( ∀ p 2 )( ∀ p 3 )( ∀ p 4 )( ∀ ℓ ) � ( I ( p 1 , ℓ ) ∧ I ( p 2 , ℓ ) ∧ I ( p 3 , ℓ ) ∧ I ( p 4 , ℓ )) → ( p 1 = p 2 ∨ p 1 = p 3 ∨ p 1 = p 4 ∨ p 2 = p 3 ∨ p 2 = p 4 ∨ p 3 = p 4 ) To say that there are infinitely many points and lines, add an infinite list of axioms • ( ∃ x 1 )( ∃ x 2 )( x 1 � = x 2 ) • ( ∃ x 1 )( ∃ x 2 )( ∃ x 3 )( x 1 � = x 2 ∨ x 1 � = x 3 ∨ x 2 � = x 3 ) • etc. G. Eric Moorhouse Semifinite Generalized Quadrangles

First-Order Axioms for Generalized Quadrangles To say that the line size is 3, add an axiom • ( ∀ p 1 )( ∀ p 2 )( ∀ p 3 )( ∀ p 4 )( ∀ ℓ ) � ( I ( p 1 , ℓ ) ∧ I ( p 2 , ℓ ) ∧ I ( p 3 , ℓ ) ∧ I ( p 4 , ℓ )) → ( p 1 = p 2 ∨ p 1 = p 3 ∨ p 1 = p 4 ∨ p 2 = p 3 ∨ p 2 = p 4 ∨ p 3 = p 4 ) To say that there are infinitely many points and lines, add an infinite list of axioms • ( ∃ x 1 )( ∃ x 2 )( x 1 � = x 2 ) • ( ∃ x 1 )( ∃ x 2 )( ∃ x 3 )( x 1 � = x 2 ∨ x 1 � = x 3 ∨ x 2 � = x 3 ) • etc. G. Eric Moorhouse Semifinite Generalized Quadrangles