Modular lattices and von Neumann regular rings
Contents
1. unique line such that so write p c instead of pe The Pasch Axiom Modular lattices and von Neumann regular rings Projective geometries A is a triple p q r of distinct points such that p qr q pr and r pq The Pasch Axiom Modular lattices and von Neumann regular rings Projective geometries A is a triple p q r of distinct points such that p qr q 6 pr and r pq The Pasch Axiom The Pasch Axiom Modular lattices and von Neumann regular rings Projective geometries A is a triple p q r of distinct points such that p qr q 6 pr and r pq The Pasch Axiom For each triangle p r The Pasch Axiom Modular lattices and von Neumann regular rings Projective geometries A is a triple p q r of distinct points such that p qr q 6 pr and r pq The Pasch Axiom For each triangle p for all distinct x pq and y ar The Pasch Axiom Modular lattices and von Neumann regular rings Projective geometries A is a triple p q r of distinct points such that p qr q pr and r pq The Pasch Axiom For each triangle p for all distinct x pq and y ar xy n pr 2 2 The Pasch Axiom Modular lattices and von Neumann regular rings Projective geometries A is a triple p q r of di2. lattices and von Neumann regular rings Desargues Illustrating Desargues Rule Modular lattices and von Neumann regular rings Desargues The Arguesian identity Modular lattices and von Neumann Desargues identity M S regular rings Desargues Hees The Arguesian identity 8 lattices and von Neumann BIB Desargues identity M Sch tzenberger 1945 B J nsson 1953 regular rings Set zo V x2 V y Al e xo V x2 Yo V y2 Desargues iem Xo V x1 V 7 The Arguesian identity Modular lattices and von Neumann regular rings Desargues identity M Sch tzenberger 1945 B Jonsson 1953 Set Zo a V x2 A a z xo V x2 A yo V ya z2 xo Voi yo V y zu UM ZB Desargues is the lattice theoretical identity The Arguesian identity Modular lattices and von Neumann BIB Desargues identity M Sch tzenberger 1945 B J nsson 1953 regular rings Set Zo V x2 V y Al e Xo V x2 Yo V y2 E Desargues 5 iem Xo V x1 yo V z m ZZ is the lattice theoretical identity xo V yo A Ga V a x2 V yo xo zV x1 V yo zV y1 The Arguesian identity Modular lattices and von Neumann BIB Desargues identity M Sch tzenberger 1945 B J nsson 1953 regular rings Set Zo
3. Modular lattices and von Neumann regular rings Most basic open problems are still unsolved c lar la Applications Ur ul i Open problems Modular lattices and von Neumann regular rings Most basic open problems are still unsolved For example Applications Open problems Modular lattices and von Neumann regular rings Most basic open problems are still unsolved Geome For example lattices 3 Problem P S CMLs Applications c ul iy Open problems Modular lattices and von Neumann regular rings Most basic open problems are still unsolved For example Problem If a lattice L embeds into some CML is this also the case for all homomorphic images of L Applications Another problem Modular lattices and von Neumann regular rings The following problem has a strong lattice theoretical content Applications Another problem Modular lattices and The following problem has a strong lattice theoretical content regular rings Problem Separativity Conjecture K R Goodearl 1995 Applications a ey Another problem lattices and von Neumann regular rings The following problem has a strong lattice theoretical content Problem Separativity Conjecture K R Goodearl 1995 Let R be a unital regular ring Applications Another problem Modular lattices and vo
4. least subspace Z such that X UY C Z The structure Sub P V the of P is a Modularity of Sub P Modular lattices and une SUME Lattice Theory regular rings Geomodular lattices Modularity of Sub P Modular lattices and ue MEE Lattice Theory regular rings is the study of all structures L V Geomodular lattices Modularity of Sub P Modular latti d ELS UM Lattice Theory regular rings is the study of all structures L V where L is a nonempty set and Geomodular lattices Modularity of Sub P Modular latti id E ELS Lattice Theory regular rings is the study of all structures L V where L is a nonempty set and V resp is the join operation resp meet operation with respect to a necessarily unique partial ordering of L Geomodular lattices Modularity of Sub P Modular latti id E ELS Lattice Theory regular rings is the study of all structures L V where L is a nonempty set and V resp is the join operation resp meet operation with respect to a necessarily unique partial predia ordering of L lattices In particular Sub P is a lattice Modularity of Sub P Modular lattices and ue SUME lattice Theory regular rings is the study of all structures L V where L is a nonempty set and V resp is the join operation resp meet operation with respect to a necessaril
5. P any singleton p and any line Ple are subspaces Sub P X X subspace of P partially ordered under C Any of subspaces is a subspace In particular for any subspaces X and Y of P one can define Projective subspaces Modular lattices and von Neumann regular rings A subset X C P is a projective of P if Vp q X pq X In particular P any singleton p and any line Ple are subspaces Sub P X X subspace of P partially ordered under C Any of subspaces is a subspace In particular for any subspaces X and Y of P one can define AAY XN0AY Projective subspaces Modular lattices and von Neumann regular rings A subset X C P is a projective of P if Vp q X pq X In particular P any singleton p and any line Ple are subspaces Sub P X X subspace of P partially ordered under C Any of subspaces is a subspace In particular for any subspaces X and Y of P one can define XAYT XN0AY X V Y join least subspace Z such that X U Y C Z Projective subspaces 8 lattices and von Neumann regular rings A subset X C P is a projective of P if Vp q X pq X In particular P any singleton p and any line Pe are subspaces Sub P X X subspace of P partially ordered under C Any of subspaces is a subspace In particular for any subspaces X and Y of P one can define XAYT QeeXxny X V Y join
6. V x2 V y Al e xo V x2 Yo V y2 E Desargues 5 iem Xo V x1 yo V z m ZZ is the lattice theoretical identity xo V yo Ga V a x2 V yo xo A zV x1 V A zV y1 A lattice is if it satisfies Desargues identity The Arguesian identity Modular lattices and von Neumann regular rings Desargues identity M Sch tzenberger 1945 B Jonsson 1953 Set Zo x V x2 A a V y z xo V x2 A yo V y2 2 x Voi yo V Za m Ala a Desargues is the lattice theoretical identity xo V yo A Ca V ya Ge V yo Go z v a V vo zv yi A lattice is if it satisfies Desargues identity Every Arguesian lattice is modular but the converse is false Desargues Rule versus Desargues identity Modular lattices and von Neumann regular rings Theorem M Sch tzenberger 1945 B J nsson 1953 Desargues Coord P S a a Desargues Rule versus Desargues identity Modular lattices and von Neumann regular rings Theorem M Sch tzenberger 1945 B J nsson 1953 A geomodular lattice is Arguesian if and only if its associated projective geometry satisfies Desargues Rule Desargues Desargues Rule versus Desargues identity Modular lattices and von Neumann regular rings Theorem M Sch tzenberger 1945 B J nsson 1953 A geomodular lattice is Arguesian if and only if its associated projective
7. Modular lattices and von Neumann regular rings Modular lattices and von Neumann regular rings Friedrich Wehrung Universit de Caen LMNO UMR 6139 D partement de Math matiques 14032 Caen cedex E mail wehrung math unicaen fr URL http www math unicaen fr wehrung Darmstadt 2008 Background projective geometries Modular lattices and von Neumann regular rings A is a structure P L where both P points and L lines are sets and e CP x L Projective geometries Background projective geometries Modular lattices and von Neumann regular rings A is a structure P L where both P points and L lines are sets and e C P x L write pe Projective pronounced contains p instead of p c geometries Background projective geometries Modular lattices and von Neumann regular rings A is a structure P L where both P points and L lines are sets and e C P x L write pe Projective pronounced contains p instead of p and the ae following axioms are satisfied Background projective geometries Modular lattices and von Neumann regular rings A is a structure P L where both P points and L lines are sets and e C P x L write pe Projective pronounced contains p instead of p and the pp following axioms are satisfied P1 every line contains at least two dist
8. Theorem R Freese 1979 There exists a lattice identity that holds in all lattices but not in every lattice Analogue for the class of all lattices does not hold Improved later by C Herrmann Applications Applications to lattice theoretical problems Modular lattices and von Neumann regular rings Theorem R Freese 1979 There exists a lattice identity that holds in all lattices but not in every lattice Analogue for the class of all lattices does not hold Improved later by C Herrmann Theorem C Herrmann 1984 Applications Applications to lattice theoretical problems Modular lattices and von Neumann regular rings Theorem R Freese 1979 There exists a lattice identity that holds in all lattices but not in every lattice Analogue for the class of all lattices does not hold Improved later by C Herrmann Theorem C Herrmann 1984 Applications m There exists a lattice identity that holds in all but not in every Arguesian lattice Applications to lattice theoretical problems 8 lattices and von Neumann regular rings Theorem R Freese 1979 There exists a lattice identity that holds in all lattices but not in every lattice Analogue for the class of all lattices does not hold Improved later by C Herrmann Theorem C Herrmann 1984 Applications m here exists a lattice identity that holds in all but not in every Arguesia
9. L such that Projective subspace lattices geomodular lattices Modular In fact Sub P satisfies much more than modularity it is abbreviation for and modular that is Ps 4 and modular Geometric lattices are often called lattices crom dular Theorem A lattice is geomodular if and only if it is isomorphic to Sub P for some projective geometry P Theorem G Birkhoff 1935 Every geomodular lattice L is that is for each x L there exists y L such that x V y 1 largest element of L and Projective subspace lattices geomodular lattices Modular In fact Sub P satisfies much more than modularity it is abbreviation for and modular that is Ps 4 and modular Geometric lattices are often called lattices crom dular Theorem A lattice is geomodular if and only if it is isomorphic to Sub P for some projective geometry P Theorem G Birkhoff 1935 Every geomodular lattice L is that is for each x L there exists y L such that x V y 1 largest element of L and x y 0 smallest element of L Projective subspace lattices geomodular lattices Modular In fact Sub P satisfies much more than modularity it is abbreviation for and modular that is Ps 4 and modular Geometric lattices are often called lattices crom dular Theorem A lattice is geomodular if and only if it is isomorphic
10. xVz Ay Vz x y V x z 2x y V x z Each of these identities defining modularity is called The modular identity 8 lattices and von Neumann regular rings Setting x x V z resp z x Z we get two equivalent forms of the modular law formulated as Geomodular lattices xVz A yVz 2 xVz Ay Vz x y V x z 2x y V x z Each of these identities defining modularity is called the The modular identity 8 lattices and von Neumann regular rings Setting x x V z resp z x Z we get two equivalent forms of the modular law formulated as Geomodular lattices xVz A yVz 2 xVz Ay Vz x y V x z 2x y V x z Each of these identities defining modularity is called the A lattice L is modular if and only if it does not contain a lattice copy of the lattice below The modular identity Modular lattices and von Neumann regular rings Setting x x V z resp z x z we get two equivalent forms of the modular law formulated as xv z y V z xv z y vz xX y V x z 2x y V x z Geomodular lattices Each of these identities defining modularity is called the A lattice L is modular if and only if it does not contain a lattice copy of the lattice Ns below Modular lattices and von Neumann regular rings Geomodular lattices Projective subspace lattices geomodular lattices In fact Sub P
11. Modular lattices and von Neumann regular rings The Coordinatization Theorem for projective geometries Von eom Staudt 19th Century O Veblen and W H Young 1910 von Neumann 1936 Coord IPS CMLs i 5 The Coordinatization Theorem for projective geometries Modular lattices and von Neumann regular rings The Coordinatization Theorem for projective geometries Von Staudt 19th Century O Veblen and W H Young 1910 von Neumann 1936 Coord 5 Every geomodular lattice is isomorphic to a product Li where each L is isomorphic to one of the types 1 3 above The Coordinatization Theorem for projective geometries Modular lattices and von Neumann regular rings The Coordinatization Theorem for projective geometries Von Staudt 19th Century O Veblen and W H Young 1910 von Neumann 1936 Coord 5 Every geomodular lattice is isomorphic to a product Li where each L is isomorphic to one of the types 1 3 above The decomposition above is unique Modular lattices and Frink s Embedding Theorem regular rings Complemented modular lattice CML CMLs Du AG Frink s Embedding Theorem Modular Hie cud Complemented modular lattice CML Modular lattice von Neumann EXE with 0 1 and Vx 3y x y 1 CMLs Frink s Embedding Theorem Modular Hi ud Complemented modular lattice CML Modular lattice von Neumann EXE
12. of all coordinatizable CMLs is FW 2006 Von Neumann s condition requires the lattice have a unit while J nsson s does not Nevertheless J nsson s Coordinatization Theorem is stated for lattices For sectionally complemented modular without unit J nsson s result extends to the case B J nsson 1962 but to the general case FW 2008 counterexample of cardinality The proof of the latter counterexample involves first used in 1957 in the theory of totally ordered abelian groups and P Gillibert and FW 2008 a tool of categorical nature Applications to lattice theoretical problems Modular lattices and von Neumann regular rings Applications Applications to lattice theoretical problems Modular lattices and von Neumann regular rings Theorem R Freese 1979 Applications Applications to lattice theoretical problems Modular lattices and von Neumann regular rings Theorem R Freese 1979 There exists a lattice identity that holds in all lattices but not in every lattice Applications Applications to lattice theoretical problems Modular lattices and von Neumann regular rings Theorem R Freese 1979 There exists a lattice identity that holds in all lattices but not in every lattice Analogue for the class of all lattices does not hold Applications Applications to lattice theoretical problems Modular lattices and von Neumann regular rings
13. satisfies much more than modularity ul i DAE Projective subspace lattices geomodular lattices lattices and icf ity it i so NEAN In fact Sub P satisfies much more than modularity it is regular rings abbreviation for and modular Geomodular lattices Projective subspace lattices geomodular lattices MEUM In fact Sub P satisfies much more than modularity it is regular rings abbreviation for and modular that u n is and modular Geomodular lattices Projective subspace lattices geomodular lattices In fact Sub P satisfies much more than modularity it is d abbreviation for and modular that is PH and modular Geometric lattices are often called lattices Geomodular lattices Projective subspace lattices geomodular lattices MAGEN In fact Sub P satisfies much more than modularity it is abbreviation for and modular that is oe and modular Geometric lattices are often called lattices Geomodular lattices Projective subspace lattices geomodular lattices Modular In fact Sub P satisfies much more than modularity it is abbreviation for and modular that is oe and modular Geometric lattices are often called lattices Geomodular lattices A lattice is geomodular if and only if it is isomorphic to Sub P fo
14. to Sub P for some projective geometry P Theorem G Birkhoff 1935 Every geomodular lattice L is that is for each x L there exists y L such that x V y 1 largest element of L and x y 0 smallest element of L Abbreviated Xx Q y 1 and we say that y isa of x Desargues Rule lattices and von Neumann Two triangles ao a1 82 and bo b1 b2 are if Desargues Desargues Rule Modular lattices and von Neumann Two triangles ao a1 82 and bo b1 b2 are if a aj A bi bj for all i Aj and Desargues Desargues Rule Modular lattices and von Neumann Two triangles ao a1 82 and bo b1 b2 are if a aj A bi bj for all i Aj and for some point p all points a bj p are collinear i e on the same line Desargues Desargues Rule Modular lattices and von Neumann Two triangles ao a1 a2 and bo b1 b are if a aj A bi bj for all i Aj and for some point p all points a bj p are collinear i e on the same line Desargues We say that ao a1 a2 and bo bi b2 are if Desargues Rule Modular lattices and von Neumann Two triangles ao a1 a2 and bo b1 b are if a aj A bi bj for all i Aj and for some point p all points a bj p are collinear i e on the same line Desargues We say that ao a1 a2 and bo bi b2 are if the points co c1 and c are
15. von Neumann regular rings Defin tion all 1 ul u a Von Neumann regular rings Modular lattices and von Neumann A ring associative not necessarily unital R is in von Neumann s sense if it satisfies CMLs Von Neumann regular rings Modular lattices and von Neumann regular rings Definition A ring associative not necessarily unital R is in von Neumann s sense if it satisfies vx GI xx x CMLs Von Neumann regular rings Modular lattices and von Neumann regular rings Definition A ring associative not necessarily unital R is in von Neumann s sense if it satisfies Yx ay xyx x CMLs the endomorphism ring of a vector space or even a semisimple module is regular Von Neumann regular rings Modular lattices and von Neumann regular rings Definition A ring associative not necessarily unital R is in von Neumann s sense if it satisfies Yx ay xyx x CMLs the endomorphism ring of a vector space or even a semisimple module is regular One can then prove that L R xR x R is a sublattice of the lattice Id Rp of all right ideals of R Von Neumann regular rings Modular lattices and von Neumann regular rings Definition A ring associative not necessarily unital R is in von Neumann s sense if it satisfies Yx ay xyx x CMLs the endomorphism ring of a vector space or ev
16. with 0 1 and Vx 3y x 8 y 1 Frink s Embedding Theorem O Frink 1946 CMLs Frink s Embedding Theorem Modular Hi ud Complemented modular lattice CML Modular lattice von Neumann XL with 0 1 and Vx 3y x 8 y 1 Frink s Embedding Theorem O Frink 1946 Every CML L embeds into some geomodular lattice L CMLs Frink s Embedding Theorem 8 lattices and Nace Complemented modular lattice CML Modular lattice XL with 0 1 and Vx 3y x 8 y 1 Frink s Embedding Theorem O Frink 1946 Every CML L embeds into some geomodular lattice L with the same 0 and 1 as L CMLs 8 lattices and von Neumann regular rings CMLs Frink s Embeddin g Theorem Complemented modular lattice CML Modular lattice with 0 1 and Vx 3 y x e y 1 Frink s Embedding T heorem O Frink 1946 Every CML L embeds into some geomodular lattice L with the same 0 and 1 as L Furthermore one can assume that L satisfies the same lattice theoretical ide ntities as L B Jonsson 1954 Frink s Embedding Theorem 8 cud Complemented modular lattice CML Modular lattice von Neumann XL with 0 1 and Vx 3y x 8 y 1 Frink s Embedding Theorem O Frink 1946 Every CML L embeds into some geomodular lattice L with the same 0 and 1 as L Furthermore one can assume that L satisfies the same CMLs lattice
17. as a spanning n frame with n gt 4 then it is coordinatizable Improved by B J nsson in 1960 J nsson s Coordinatization Theorem If a CML has a large 4 frame or it is Arguesian and it has a large 3 frame then it is coordinatizable CMLs A much more transparent proof of J nsson s Coordinatization Theorem has recently been found by C Herrmann Coordinatization of CMLs cont d Modular ao lattices and Both von Neumann s condition and J nsson s condition can be von Neumann regular rings expressed by first order axioms Nevertheless CMLs Coordinatization of CMLs cont d Modular lattices and Both von Neumann s condition and J nsson s condition can be von eumann regular rings expressed by first order axioms Nevertheless The class of all coordinatizable CMLs is FW 2006 CMLs Coordinatization of CMLs cont d Modular ai Both von Neumann s condition and J nsson s condition can be regular rings expressed by first order axioms Nevertheless The class of all coordinatizable CMLs is FW 2006 Von Neumann s condition requires the lattice have a unit while J nsson s does not CMLs Coordinatization of CMLs cont d ties and Both von Neumann s condition and J nsson s condition can be regular rings expressed by first order axioms Nevertheless The class of all coordinatizable CMLs is FW 2006 Von Neumann s condition requires the lattice have a unit while J ns
18. collinear where Desargues Rule Modular lattices and von Neumann regular rings Two triangles ao a1 82 and bo b1 b are if a aj A bi bj for all i Aj and for some point p all points a bj p are collinear i e on the same line Desargues We say that ao a1 a2 and bo bi b2 are if the points co c1 and c are collinear where a1 a2 b b2 co and cyclically Desargues Rule Modular lattices and von Neumann regular rings Definition Two triangles ao a1 82 and bo b1 b2 are if a aj A bi bj for all i Aj and for some point p all points a bj p are collinear i e on the same line Desargues We say that ao a1 a2 and bo b b2 are if the points co c1 and c are collinear where a1 a2 b b2 co and cyclically We say that the projective geometry P is or satisfies if Modular lattices and von Neumann regular rings Desargues Desargues Rule Two triangles ao a1 82 and bo b1 b2 are if a aj A bi bj for all i Aj and for some point p all points a bj p are collinear i e on the same line We say that ao a1 a2 and bo bi b2 are if the points co c1 and c are collinear where a1 a2 b b2 co and cyclically We say that the projective geometry P is or satisfies if any two centrally perspective triangles are also axially perspective Illustrating Desargues Rule
19. egular rings A is a structure P L c where both P points and L lines are sets and e C P x L write pe Projective pronounced contains p instead of p and the Ere following axioms are satisfied P1 every line contains at least two distinct points P2 any two distinct points are contained in exactly one line P3 more detail later By Axioms P1 and P2 L ipeP petj Background projective geometries Modular lattices and von Neumann regular rings A is a structure P L c where both P points and L lines are sets and e C P x L write pe Projective pronounced contains p instead of p and the Ere following axioms are satisfied P1 every line contains at least two distinct points P2 any two distinct points are contained in exactly one line P3 more detail later By Axioms P1 and P2 unique line such that p qe Background projective geometries Modular lattices and von Neumann regular rings A is a structure P L c where both P points and L lines are sets and e C P x L write pe Projective pronounced contains p instead of p and the following axioms are satisfied P1 every line contains at least two distinct points P2 any two distinct points are contained in exactly one line P3 more detail later By Axioms P1 and P2
20. en a semisimple module is regular One can then prove that L R xR x R is a sublattice of the lattice Id Rg of all right ideals of R in particular it is modular Von Neumann regular rings Modular lattices and von Neumann regular rings Definition A ring associative not necessarily unital R is in von Neumann s sense if it satisfies Yx ay xyx x CMLs the endomorphism ring of a vector space or even a semisimple module is regular One can then prove that L R xR x R is a sublattice of the lattice Id Rp of all right ideals of R in particular it is modular More can be proved Coordinatizable lattices Modular lattices and von Neumann regular rings Theorem Von Neumann 1936 Fryer and Halperin 1954 Coord P S CMLs gt a I Coordinatizable lattices Modular lattices and von Neumann BIB heorem Von Neumann 1936 Fryer and Halperin 1954 regular rings The lattice L R is modular and also the latter meaning that CMLs Coordinatizable lattices 8 lattices and von Neumann fag heorem Von Neumann 1936 Fryer and Halperin 1954 regular rings The lattice L R is modular and also the latter meaning that Vx 32 ez y CMLs Coordinatizable lattices Modular lattices and von Neumann BIB heorem Von Neumann 1936 Fryer and Halperin 1954 regular rings The lattice L R
21. eumann frames Modular lattices and Definition regular rings a E E E Elements a b in a modular lattice L with 0 are notation a e b if a c b c Elements ag an are if for each k n ick An is a system a 0 i lt n c 1 i n where a 0 i lt n is independent and ao a for 1 lt i lt _n The frame is d Vienn Von Neumann frames Modular lattices and ANa Definition regular rings a E E Elements a b in a modular lattice L with 0 are notation a e b if a c bG c Elements 80 seg if Sp O for each k n ick a An is a system 4 0 i 1 i n where a 0 i lt n is independent and ao a for 1 lt i lt _n The frame is i L Vieni if every element of L is a finite join of elements perspective to parts of ao Von Neumann frames Modular lattices and ANa Definition regular rings a E E Elements a b in a modular lattice L with 0 are notation a e b if a c b c Elements ag ran are if Sp O for each k n ick a An is a system 4 0 i lt 1 i n where a 0 i lt n is independent and ao a for 1 lt i lt _n The frame is vule Vien din if every element of L is a finite join of elements perspective to parts of ag Hence spanning large Von Neumann regular rings Modular lattices and
22. geometry satisfies Desargues Rule Desargues Other classes of Arguesian lattices Desargues Rule versus Desargues identity Modular lattices and von Neumann regular rings Theorem M Sch tzenberger 1945 B J nsson 1953 A geomodular lattice is Arguesian if and only if its associated projective geometry satisfies Desargues Rule Desargues Other classes of Arguesian lattices m The normal subgroup lattice NSub G of any group G Desargues Rule versus Desargues identity Modular lattices and von Neumann regular rings Theorem M Sch tzenberger 1945 B J nsson 1953 A geomodular lattice is Arguesian if and only if its associated projective geometry satisfies Desargues Rule Desargues Other classes of Arguesian lattices m The normal subgroup lattice NSub G of any group G m The submodule lattice Sub M of any module M Desargues Rule versus Desargues identity Modular lattices and von Neumann regular rings Theorem M Sch tzenberger 1945 B J nsson 1953 A geomodular lattice is Arguesian if and only if its associated projective geometry satisfies Desargues Rule Desargues Other classes of Arguesian lattices m The normal subgroup lattice NSub G of any group G m The submodule lattice Sub M of any module M m more general Any lattice of permuting equivalence relations on a given set Note Arguesian is then not the end of the story projective spaces Modula
23. inct points Background projective geometries Modular lattices and von Neumann regular rings A is a structure P L c where both P points and L lines are sets and e C P x L write pe Projective pronounced contains p instead of p and the Ere following axioms are satisfied P1 every line contains at least two distinct points P2 any two distinct points are contained in exactly one line Background projective geometries Modular lattices and von Neumann regular rings A is a structure P L c where both P points and L lines are sets and e C P x L write pe Projective pronounced contains p instead of p and the following axioms are satisfied P1 every line contains at least two distinct points P2 any two distinct points are contained in exactly one line P3 more detail later Background projective geometries Modular lattices and von Neumann regular rings A is a structure P L c where both P points and L lines are sets and e C P x L write pe Projective pronounced contains p instead of p and the Ere following axioms are satisfied P1 every line contains at least two distinct points P2 any two distinct points are contained in exactly one line P3 more detail later By Axioms P1 and P2 Background projective geometries Modular lattices and von Neumann r
24. is modular and also the latter meaning that Vx 32 ez y In particular IL R is complemented modular if and only if R is unital For modular lattices CMLs Coordinatizable lattices Modular lattices and von Neumann fag heorem Von Neumann 1936 Fryer and Halperin 1954 regular rings The lattice L R is modular and also the latter meaning that Vx y 3 z x ez y In particular IL R is complemented modular if and only if R CMLs is unital For modular lattices ES Coordinatizable lattices Modular lattices and von Neumann fag heorem Von Neumann 1936 Fryer and Halperin 1954 regular rings The lattice L R is modular and also the latter meaning that Vx 32 ez y In particular IL R is complemented modular if and only if R Ae is unital For modular lattices gt Definition A lattice is if it is isomorphic to L R for some regular ring R Coordinatizable lattices Modular lattices and von Neumann fag heorem Von Neumann 1936 Fryer and Halperin 1954 regular rings The lattice L R is modular and also the latter meaning that Vx 32 ez y In particular IL R is complemented modular if and only if R Ae is unital For modular lattices gt Definition A lattice is if it is isomorphic to L R for some regular ring R The easiest example of
25. ition regular rings E E i k Elements a b in a modular lattice L with 0 are notation a e b if a c b c Elements ag e an T if CMLs Von Neumann frames Modular lattices and Definition regular rings E E i k Elements a b in a modular lattice L with 0 are notation a e b if a c b c Elements ag If AN es for each k n ick CMLs Von Neumann frames Modular lattices and Definition regular rings 2 E E E Elements a b in a modular lattice L with 0 are notation a e b if a c b c Elements ag rane are if for each k n ick us An is a system 4 0 i lt n c 1 i n Von Neumann frames Modular lattices and Definition regular rings a E E E Elements a b in a modular lattice L with 0 are notation a e b if a c b c Elements ag an are if for each k n ick An is a system 4 0 i n c 1 i n where a 0 i lt n is independent and ao a for 1l lt i lt n Von Neumann frames Modular lattices and Definition regular rings a E E E Elements a b in a modular lattice L with 0 are notation a e b if a c b c Elements ag an are if for each k n ick An is a system a 0 i lt n c 1 i n where a 0 i lt n is independent and ao a for 1 lt i lt _n The frame is Von N
26. morphism types of finitely generated projective right R modules Is V R that is does it satisfy the following statement Applications Vx y 2x 2y x y E y The problem above is also open for C algebras of real rank zero and even for general Warfield lattices and von Neumann regular rings Applications Variety is the spice of life A is the class of all structures here lattices that satisfy a given set of identities Variety is the spice of life lattices and von Neumann regular rings A is the class of all structures here lattices that satisfy a given set of identities For example is the variety of all lattices M is the variety of all modular lattices M5 is the variety generated by Ns Applications Variety is the spice of life lattices and von Neumann regular rings A is the class of all structures here lattices that satisfy a given set of identities For example is the variety of all lattices M is the variety of all modular lattices M5 is the variety generated by s Partial picture of the lattice of all varieties of lattices Applications Variety is the spice of life lattices and A is the class of all structures here lattices that png satisfy a given set of identities For example is the variety of all lattices M is the variety of all modular lattices M5 is the variety generated by s Partial pictu
27. n Neumann regular rings The following problem has a strong lattice theoretical content Problem Separativity Conjecture K R Goodearl 1995 Let R be a unital regular ring Denote by V R the commutative monoid of all isomorphism types of finitely generated projective right R modules Applications Another problem Modular lattices and von Neumann regular rings The following problem has a strong lattice theoretical content Problem Separativity Conjecture K R Goodearl 1995 Let R be a unital regular ring Denote by V R the commutative monoid of all isomorphism types of finitely generated projective right R modules Is V R that is does it satisfy the following statement Applications Another problem Modular lattices and ewe The following problem has a strong lattice theoretical content regular rings Problem Separativity Conjecture K R Goodearl 1995 Let R be a unital regular ring Denote by V R the commutative monoid of all isomorphism types of finitely generated projective right R modules Is V R that is does it satisfy the following statement Vx y 2x dy x y x y Applications Another problem Modular lattices and ewe The following problem has a strong lattice theoretical content regular rings Problem Separativity Conjecture K R Goodearl 1995 Let R be a unital regular ring Denote by V R the commutative monoid of all iso
28. n lattice m he set of all identities satisfied by all lattices is not generated by any finite subset Word problem for modular lattices Modular lattices and von Neumann regular rings Theorem C Herrmann 1983 P attices 2S Ch Applications a I at iy Word problem for modular lattices Modular lattices and von Neumann regular rings Theorem C Herrmann 1983 The word problem for free modular lattices on four generators is recursively unsolvable Applications Word problem for modular lattices Modular lattices and von Neumann regular rings Theorem C Herrmann 1983 The word problem for free modular lattices on four generators is recursively unsolvable The corresponding statement with five instead of four was proved by R Freese in 1980 Applications Word problem for modular lattices Modular lattices and von Neumann regular rings Theorem C Herrmann 1983 The word problem for free modular lattices on four generators is recursively unsolvable The corresponding statement with five instead of four was proved by R Freese in 1980 The free modular lattice on three generators is finite with 28 elements R Dedekind 1900 so one can t go down to three Applications Word problem for modular lattices 8 lattices and von Neumann regular rings Theorem C Herrmann 1983 The word problem for free modula
29. non coordinatizable CML is M7 Coordinatization of CMLs Modular lattices and von Neumann regular rings Neumann s Coordinatization Theore B A a ey ul iy FAG Coordinatization of CMLs Modular lattices and von Neumann regular rings Von Neumann s Coordinatization Theorem If a CML has a spanning n frame with n gt 4 then it is coordinatizable CMLs Coordinatization of CMLs Modular lattices and von Neumann regular rings Von Neumann s Coordinatization Theorem If a CML has a spanning n frame with n gt 4 then it is coordinatizable Improved by B J nsson in 1960 CMLs Coordinatization of CMLs Modular lattices and von Neumann regular rings Von Neumann s Coordinatization Theorem If a CML has a spanning n frame with n gt 4 then it is coordinatizable Improved by B J nsson in 1960 J nsson s Coordinatization Theorem CMLs Coordinatization of CMLs Modular lattices and von Neumann regular rings Von Neumann s Coordinatization Theorem If a CML has a spanning n frame with n gt 4 then it is coordinatizable Improved by B J nsson in 1960 J nsson s Coordinatization Theorem If a CML has a large 4 frame or it is Arguesian and it has a large 3 frame then it is coordinatizable CMLs Coordinatization of CMLs Modular lattices and von Neumann regular rings Von Neumann s Coordinatization Theorem If a CML h
30. p q X Projective geometries Projective subspaces lattices and von Neumann A subset X C P is a projective of P if Vb q X T pq X Projective geometries Projective subspaces lattices and von Neumann Aun A subset X C P is a projective of P if Vp q X pq X In particular P any singleton p and any line Projective geometries are su bspaces Projective subspaces lattices and von Neumann regular rings A subset X C P is a projective of P if Vp q X pq X In particular P any singleton p and any line De are subspaces Sub P X X subspace of P Projective subspaces lattices and von Neumann regular rings A subset X C P is a projective of P if Vp q X pq X In particular P any singleton p and any line De are subspaces Sub P X X subspace of P partially ordered under C Projective subspaces Modular lattices and von Neumann regular rings A subset X C P is a projective of P if Vp q X pq X In particular P any singleton p and any line Ple are subspaces Sub P X X subspace of P partially ordered under C Any of subspaces is a subspace Projective subspaces Modular lattices and von Neumann A subset X C P is a projective of P if Vp q X pq X In particular
31. r lattices and von Neumann regular rings Fundamental examples of geomodular lattices Coord P S CMLs 1 The two element lattice 2 0 1 ul i AAA Fundamental examples of geomodular lattices projective spaces Modular lattices and von Neumann regular rings 1 The two element lattice 2 0 1 the lattice of length two and atoms for a cardinal Coord P S Fundamental examples of geomodular lattices projective spaces Modular lattices and von Neumann regular rings 1 The two element lattice 2 0 1 the lattice of length two and Mur for a cardinal 1 Coord P S 2 Fundamental examples of geomodular lattices projective spaces Modular lattices and von Neumann regular rings 1 The two element lattice 2 0 1 the lattice M of length two and atoms for a cardinal 1 Coord P S 2 2 the lattice Sub V of all subspaces of a vector space V of dimension gt 3 over any division ring Fundamental examples of geomodular lattices projective spaces Modular lattices and von Neumann regular rings 1 The two element lattice 2 0 1 the lattice M of length two and atoms for a cardinal 1 Coord 2 2 the lattice Sub V of all subspaces of a vector space V of dimension gt 3 over any division ring 5 The Coordinatization Theorem for projective geometries
32. r lattices on four generators is recursively unsolvable The corresponding statement with five instead of four was proved by R Freese in 1980 The free modular lattice on three generators is finite with 28 elements R Dedekind 1900 so one can t go down to three Applications Remark Word problem for modular lattices Modular lattices and von Neumann regular rings Theorem C Herrmann 1983 The word problem for free modular lattices on four generators is recursively unsolvable The corresponding statement with five instead of four was proved by R Freese in 1980 The free modular lattice on three generators is finite with 28 elements R Dedekind 1900 so one can t go down to three Applications Remark m he word problem for all lattices is solvable in polynomial time Word problem for modular lattices Modular lattices and von Neumann regular rings Theorem C Herrmann 1983 The word problem for free modular lattices on four generators is recursively unsolvable The corresponding statement with five instead of four was proved by R Freese in 1980 The free modular lattice on three generators is finite with 28 elements R Dedekind 1900 so one can t go down to three Applications Remark m he word problem for all lattices is solvable in polynomial time m he word problem for all lattices is NP complete Open problems
33. r some projective geometry P Modular lattices and von Neumann regular rings Geomodular lattices Projective subspace lattices geomodular lattices In fact Sub P satisfies much more than modularity it is abbreviation for and modular that is Pu and modular Geometric lattices are often called lattices Theorem A lattice is geomodular if and only if it is isomorphic to Sub P for some projective geometry P Theorem G Birkhoff 1935 Modular lattices and von Neumann regular rings Geomodular lattices Projective subspace lattices geomodular lattices In fact Sub P satisfies much more than modularity it is abbreviation for and modular that is Pu and modular Geometric lattices are often called lattices Theorem A lattice is geomodular if and only if it is isomorphic to Sub P for some projective geometry P Theorem G Birkhoff 1935 Every geomodular lattice L is Projective subspace lattices geomodular lattices Modular In fact Sub P satisfies much more than modularity it is abbreviation for and modular that is Ps 4 and modular Geometric lattices are often called lattices Geomodular Theorem lattices A lattice is geomodular if and only if it is isomorphic to Sub P for some projective geometry P Theorem G Birkhoff 1935 Every geomodular lattice L is that is for each x L there exists y
34. re of the lattice of all varieties of lattices Applications
35. son s does not Nevertheless J nsson s Coordinatization Theorem is stated for lattices CMLs Modular lattices and von Neumann regular rings CMLs Coordinatization of CMLs cont d Both von Neumann s condition and J nsson s condition can be expressed by first order axioms Nevertheless The class of all coordinatizable CMLs is FW 2006 Von Neumann s condition requires the lattice have a unit while J nsson s does not Nevertheless J nsson s Coordinatization Theorem is stated for lattices For sectionally complemented modular latices without unit J nsson s result extends to the case B J nsson 1962 Modular lattices and von Neumann regular rings CMLs Coordinatization of CMLs cont d Both von Neumann s condition and J nsson s condition can be expressed by first order axioms Nevertheless The class of all coordinatizable CMLs is FW 2006 Von Neumann s condition requires the lattice have a unit while J nsson s does not Nevertheless J nsson s Coordinatization Theorem is stated for lattices For sectionally complemented modular lattites without unit J nsson s result extends to the case B J nsson 1962 but to the general case FW 2008 counterexample of cardinality Modular lattices and von Neumann regular rings CMLs Coordinatization of CMLs cont d Both von Neumann s condition and J nsson s condition can be expressed by first order axioms Nevertheless The class
36. stinct points such that p qr q pr and r pq The Pasch Axiom For each triangle p for all distinct x pq and y qr xy pr Z 9 There are no parallels The Pasch Axiom Modular lattices and von Neumann A is a triple p q r of distinct points such that p ar q pr and r pq The Pasch Axiom For each triangle p q r for all distinct x pq and y qr xy pr Z 9 There are no parallels Projective geometries q The Pasch Axiom Modular lattices and von Neumann A is a triple p q r of distinct points such that p ar q pr and r pq The Pasch Axiom For each triangle p q r for all distinct x pq and y qr xy pr Z 89 There are no parallels Projective geometries q The Pasch Axiom Modular lattices and von Neumann A is a triple p q r of distinct points such that p ar q pr and r pq The Pasch Axiom For each triangle p q r for all distinct x pq and y qr xy pr Z 9 There are no parallels Projective geometries q Projective subspaces lattices and von Neumann A subset X C P is a projective of P if Projective geometries Projective subspaces lattices and von Neumann A subset X C P is a projective of P if
37. theoretical identities as L B J nsson 1954 e g the Arguesian identity Frink s Embedding Theorem Modular ee cud Complemented modular lattice CML Modular lattice von Neumann EXE with 0 1 and Vx 3y x 8 y 1 Frink s Embedding Theorem O Frink 1946 Every CML L embeds into some geomodular lattice L with the same 0 and 1 as L Furthermore one can assume that L satisfies the same CMLs lattice theoretical identities as L B J nsson 1954 e g the Arguesian identity C Herrmann and A Huhn 1975 Frink s Embedding Theorem Modular ee cud Complemented modular lattice CML Modular lattice von Neumann EXE with 0 1 and Vx 3y x 8 y 1 Frink s Embedding Theorem O Frink 1946 Every CML L embeds into some geomodular lattice L with the same 0 and 1 as L Furthermore one can assume that L satisfies the same CMLs lattice theoretical identities as L B J nsson 1954 e g the Arguesian identity C Herrmann and A Huhn 1975 Sub Z AZ the subgroup lattice of Z 4Z Von Neumann frames Modular lattices and von Neumann regular rings P ct geometries Geor la lattic D oord P S 8 Von Neumann frames Modular lattices and ANa Definition regular rings E E i Elements a b in a modular lattice L with 0 are notation a e b if aG c boc CMLs Von Neumann frames Modular lattices and Defin
38. very special sort of lattice The lattice Sub P is that is it satisfies the rule Vy V z the Modular lattices and von Neumann regular rings Geomodular lattices The modular identity Setting x x V z resp z x z 5 ul i AG The modular identity 8 lattices and von Neumann regular rings Setting x x V z resp z X z we get two equivalent forms of the modular law Geomodular lattices The modular identity 8 lattices and von Neumann regular rings Setting x x V z resp z X z we get two equivalent forms of the modular law formulated as Geomodular lattices The modular identity 8 lattices and von Neumann regular rings Setting x x V z resp z x Z we get two equivalent forms of the modular law formulated as Geomodular lattices xVz A yVz x v z y Vz The modular identity 8 lattices and von Neumann regular rings Setting x x V z resp z x Z we get two equivalent forms of the modular law formulated as Geomodular lattices xVz A yVz 2 x V z y Vz x y V x z 2x y V x z The modular identity 8 lattices and von Neumann regular rings Setting x x V z resp z x Z we get two equivalent forms of the modular law formulated as Geomodular lattices xVz A yVz 2
39. y unique partial ordering of L Geomodular lattices In particular Sub P is a lattice It is in fact a very special sort of lattice Lemma Modularity of Sub P Modular lattices and ue SUME lattice Theory regular rings is the study of all structures L V where L is a nonempty set and V resp is the join operation resp meet operation with respect to a necessarily unique partial ordering of L Geomodular lattices In particular Sub P is a lattice It is in fact a very special sort of lattice Lemma The lattice Sub P is that is it satisfies the rule Modularity of Sub P Modular lattices and ae SUME lattice Theory regular rings is the study of all structures L V where L is a nonempty set and V resp is the join operation resp meet operation with respect to a necessarily unique partial ordering of L Geomodular lattices In particular Sub P is a lattice It is in fact a very special sort of lattice The lattice Sub P is that is it satisfies the rule xe y Vive Modularity of Sub P Modular lattices and ue UNE lattice Theory regular rings is the study of all structures L V where L is a nonempty set and V resp is the join operation resp meet operation with respect to a necessarily unique partial ordering of L Geomodular lattices In particular Sub P is a lattice It is in fact a
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