Matrices in modular lattices Benedek Skublics G abor Cz edli - - PowerPoint PPT Presentation

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.

p q r y x

• 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.

p q r y x 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 Dκ

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 Dκ

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

∞ 1

Figure: F = GF(2), κ = 3

• B. S. (University of Szeged)

Matrices in modular lattices CSM2010 7 / 16

Coordinatization Addition

∞ a b l

• B. S. (University of Szeged)

Matrices in modular lattices CSM2010 8 / 16

Coordinatization Addition

∞ a b l p q

1 p, q

• B. S. (University of Szeged)

Matrices in modular lattices CSM2010 8 / 16

Coordinatization Addition

∞ a b l p q r

1 p, q 2 r = (a ∨ p) ∧ (q ∨ ∞)

• B. S. (University of Szeged)

Matrices in modular lattices CSM2010 8 / 16

Coordinatization Addition

∞ a b l p q r s

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

∞ a b l p q r s 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 b a 1

• B. S. (University of Szeged)

Matrices in modular lattices CSM2010 9 / 16

Coordinatization Multiplication

∞ l b a 1 q p

1 p, q

• B. S. (University of Szeged)

Matrices in modular lattices CSM2010 9 / 16

Coordinatization Multiplication

∞ l b a 1 q r p

1 p, q 2 r = (1 ∨ p) ∧ (q ∨ b)

• B. S. (University of Szeged)

Matrices in modular lattices CSM2010 9 / 16

Coordinatization Multiplication

∞ l b a 1 q r s p

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

∞ l b a 1 q r s ab p

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) Let L be a bounded modular lattice. For a = (a1, . . . , an) ∈ Ln and

• c = (. . . , cij, . . .) ∈ Ln(n−1) (i = j) we say that (

a, c) is a (spanning von Neumann) n-frame of L, if:

1 a1, . . . , an ≤ L is a Boolean sublattice (∼

= 2n) with atoms ai;

2 0L, 1L ∈ a1, . . . , an; 3 aj, cjk, ak is an M3 for j = k and 4 cik = cki = (cij + cjk)(ai + ak) 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