Title page Matrices in modular lattices Benedek Skublics G´ abor Cz´ edli University of Szeged Bolyai Institute The First Conference of PhD Students in Mathematics Szeged, 2010 B. S. (University of Szeged) Matrices in modular lattices CSM2010 1 / 16
Title page The outline of the talk: coordinatization von Neumann n -frames B. S. (University of Szeged) Matrices in modular lattices CSM2010 2 / 16
Coordinatization Definition (nondegenerate projective space) Let A be a set (”points”), L ⊆ P ( A ) (”lines”). Then ( A , L ) is a projective space iff the following properties hold: 1 Every l ∈ L has at least three elements. 2 ∀ distinct p , q ∈ A , ∃ ! l ∈ L satisfying p , q ∈ l . 3 The Pasch Axiom holds. B. S. (University of Szeged) Matrices in modular lattices CSM2010 3 / 16
Coordinatization Definition (nondegenerate projective space) Let A be a set (”points”), L ⊆ P ( A ) (”lines”). Then ( A , L ) is a projective space iff the following properties hold: 1 Every l ∈ L has at least three elements. 2 ∀ distinct p , q ∈ A , ∃ ! l ∈ L satisfying p , q ∈ l . 3 The Pasch Axiom holds. q x y p r B. S. (University of Szeged) Matrices in modular lattices CSM2010 3 / 16
Coordinatization Definition (nondegenerate projective space) Let A be a set (”points”), L ⊆ P ( A ) (”lines”). Then ( A , L ) is a projective space iff the following properties hold: 1 Every l ∈ L has at least three elements. 2 ∀ distinct p , q ∈ A , ∃ ! l ∈ L satisfying p , q ∈ l . 3 The Pasch Axiom holds. q x y p r z B. S. (University of Szeged) Matrices in modular lattices CSM2010 3 / 16
Coordinatization Example Let F be a field and let κ > 2 be a cardinal number. Then the one and two dimensional subspaces of F κ form a projective space. B. S. (University of Szeged) Matrices in modular lattices CSM2010 4 / 16
Coordinatization Example Let F be a field and let κ > 2 be a cardinal number. Then the one and two dimensional subspaces of F κ form a projective space. The subspaces of F κ form a modular lattice. B. S. (University of Szeged) Matrices in modular lattices CSM2010 4 / 16
Coordinatization Example Let F be a field and let κ > 2 be a cardinal number. Then the one and two dimensional subspaces of F κ form a projective space. The subspaces of F κ form a modular lattice. Modularity: x ∧ ( y ∨ ( x ∧ z )) ≤ ( x ∧ y ) ∨ ( x ∧ z ). B. S. (University of Szeged) Matrices in modular lattices CSM2010 4 / 16
Coordinatization Figure: F = GF(2) , κ = 3 B. S. (University of Szeged) Matrices in modular lattices CSM2010 5 / 16
Coordinatization Figure: F = GF(2) , κ = 3 B. S. (University of Szeged) Matrices in modular lattices CSM2010 5 / 16
Coordinatization Figure: F = GF(2) , κ = 3 B. S. (University of Szeged) Matrices in modular lattices CSM2010 5 / 16
Coordinatization Theorem (coordinatization) Let L be a directly irreducible Arguesian geometric lattice of length at least three. Then there exist a division ring D (”noncommutative field”), unique up to isomorphism, and a unique cardinal number κ such that L is isomorphic to the submodul (”subspace”) lattice of D κ D . B. S. (University of Szeged) Matrices in modular lattices CSM2010 6 / 16
Coordinatization Theorem (coordinatization) Let L be a directly irreducible Arguesian geometric lattice of length at least three. Then there exist a division ring D (”noncommutative field”), unique up to isomorphism, and a unique cardinal number κ such that L is isomorphic to the submodul (”subspace”) lattice of D κ D . l line, 0 , 1 , ∞ ∈ l distinct points. D = l − {∞} . B. S. (University of Szeged) Matrices in modular lattices CSM2010 6 / 16
Coordinatization Figure: F = GF(2) , κ = 3 B. S. (University of Szeged) Matrices in modular lattices CSM2010 7 / 16
Coordinatization 0 1 ∞ Figure: F = GF(2) , κ = 3 B. S. (University of Szeged) Matrices in modular lattices CSM2010 7 / 16
Coordinatization Addition l 0 a b ∞ B. S. (University of Szeged) Matrices in modular lattices CSM2010 8 / 16
Coordinatization Addition q p l 0 a b ∞ 1 p , q B. S. (University of Szeged) Matrices in modular lattices CSM2010 8 / 16
Coordinatization Addition r q p l 0 a b ∞ 1 p , q 2 r = ( a ∨ p ) ∧ ( q ∨ ∞ ) B. S. (University of Szeged) Matrices in modular lattices CSM2010 8 / 16
Coordinatization Addition r q p s l 0 a b ∞ 1 p , q 2 r = ( a ∨ p ) ∧ ( q ∨ ∞ ) 3 s = ( p ∨ ∞ ) ∧ ( b ∨ q ) B. S. (University of Szeged) Matrices in modular lattices CSM2010 8 / 16
Coordinatization Addition r q p s l 0 a b a+b ∞ 1 p , q 2 r = ( a ∨ p ) ∧ ( q ∨ ∞ ) 3 s = ( p ∨ ∞ ) ∧ ( b ∨ q ) 4 a + b = ( r ∨ s ) ∧ l B. S. (University of Szeged) Matrices in modular lattices CSM2010 8 / 16
Coordinatization Multiplication l 0 1 a b ∞ B. S. (University of Szeged) Matrices in modular lattices CSM2010 9 / 16
Coordinatization Multiplication p q l 0 1 a b ∞ 1 p , q B. S. (University of Szeged) Matrices in modular lattices CSM2010 9 / 16
Coordinatization Multiplication r p q l 0 1 a b ∞ 1 p , q 2 r = (1 ∨ p ) ∧ ( q ∨ b ) B. S. (University of Szeged) Matrices in modular lattices CSM2010 9 / 16
Coordinatization Multiplication r s p q l 0 1 a b ∞ 1 p , q 2 r = (1 ∨ p ) ∧ ( q ∨ b ) 3 s = (0 ∨ r ) ∧ ( p ∨ a ) B. S. (University of Szeged) Matrices in modular lattices CSM2010 9 / 16
Coordinatization Multiplication r s p q l 0 1 a b ab ∞ 1 p , q 2 r = (1 ∨ p ) ∧ ( q ∨ b ) 3 s = (0 ∨ r ) ∧ ( p ∨ a ) 4 a + b = ( q ∨ s ) ∧ l B. S. (University of Szeged) Matrices in modular lattices CSM2010 9 / 16
n -frames John von Neumann generalized the coordinatization for von Neumann regular rings ( ∀ a ∃ x : axa = a ) and complemented modular lattices containing a spanning n -frame. B. S. (University of Szeged) Matrices in modular lattices CSM2010 10 / 16
n -frames Several people have investigated n -frames. B. Artmann R. Freese A. Day and D. Pickering A. Huhn C. Herrmann G. Cz´ edli B. S. (University of Szeged) Matrices in modular lattices CSM2010 11 / 16
n -frames Definition ( n -frame) a = ( a 1 , . . . , a n ) ∈ L n and Let L be a bounded modular lattice. For � c = ( . . . , c ij , . . . ) ∈ L n ( n − 1) ( i � = j ) we say that ( � � a ,� c ) is a (spanning von Neumann) n-frame of L , if: 1 � a 1 , . . . , a n � ≤ L is a Boolean sublattice ( ∼ = 2 n ) with atoms a i ; 2 0 L , 1 L ∈ � a 1 , . . . , a n � ; 3 � a j , c jk , a k � is an M 3 for j � = k and 4 c ik = c ki = ( c ij + c jk )( a i + a k ) for distinct i , j , k . B. S. (University of Szeged) Matrices in modular lattices CSM2010 12 / 16
n -frames n -frame Figure: von Neumann 3-frame B. S. (University of Szeged) Matrices in modular lattices CSM2010 13 / 16
n -frames Product-frame Figure: Product-frame (G. Cz´ edli) B. S. (University of Szeged) Matrices in modular lattices CSM2010 14 / 16
n -frames Product-frame Theorem (G. Cz´ edli, B. S.) ”The ring of the (original) frame is the matrix ring over the ring of the product frame.” B. S. (University of Szeged) Matrices in modular lattices CSM2010 15 / 16
n -frames Product-frame Thank you! B. S. (University of Szeged) Matrices in modular lattices CSM2010 16 / 16
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