Orders on Structures and Structure of Orders Valentina Harizanov - PowerPoint PPT Presentation

CiE 2011, So…a, Bulgaria Special Session Computability in Analysis, Algebra, and Geometry Orders on Structures and Structure of Orders Valentina Harizanov Department of Mathematics George Washington University harizanv@gwu.edu http://home.gwu.edu/~harizanv/

� Magma is a nonempty set with a binary operation: ( M; � ) � A linear (partial) ordering < of the domain M is a (partial) left-order on the structure ( M; � ) if it is left invariant with respect to � : ( 8 x; y; z )[ x < y ) z � x < z � y ] � < is a bi-order ( order ) on the structure if ( 8 x; y; z )[ x < y ) z � x < z � y ^ x � z < y � z ] � LO ( M ) the set of left orders on M RO ( M ) the set of right orders on M BiO ( M ) the set of bi-orders on M

� Given a left order < l on a group G , we have a right order < r : x < r y , y � 1 < l x � 1 � G is left-orderable group ) G is torsion-free torsion-free: ( 8 x 2 G � f e g )[ order ( x ) = 1 ] e < x ) x < x 2 < � � � < x n � (Levy) G is abelian and torsion-free ) G is orderable � (Kokorin and Kopytov) Every torsion-free nilpotent group is orderable.

� Not every torsion-free group is left-orderable. � Let < be a partial left order on a group G Positive partial cone : P = f a 2 G j a � e g Negative partial cone: P � 1 = f a 2 G j a � e g 1. PP � P ( P sub-semigroup of G ) 2. P \ P � 1 = f e g ( P pure )

� P with 1 & 2 de…nes a partial left order � P on G : x � P y , x � 1 y 2 P x � P y ) x � 1 y 2 P ) x � 1 z � 1 zy = ( zx ) � 1 ( zy ) 2 P ) zx � P zy � P with 1 & 2 de…nes a left order if 3. P [ P � 1 = G ( P total )

� P with 1, 2 & 3 de…nes a bi-order if: 4. ( 8 g 2 G )[ g � 1 Pg � P ] ( P normal ) bi-order > : let g 2 G x > e ) g � 1 xg > g � 1 eg = e P normal: let x � P y , z 2 G right invariant: x � 1 y 2 P ) z � 1 x � 1 yz 2 P ( xz ) � 1 yz 2 P ) xz � P yz � For groups, orders often identi…ed with their positive cones.

� Example: G = Z � Z bi-orderable with a positive cone P = f ( a; b ) j 0 < a _ ( a = 0 ^ 0 � b ) g : � Fundamental group of Klein bottle D E x; y j xyx � 1 y = e G = left-orderable, but not bi-orderable. Positive cone P = f x n y m j n > 0 _ ( n = 0 ^ m � 0) g de…nes a left order on G . If < bi-order on G , then y > e or y < e y > e ) y � 1 = xyx � 1 > e y < e ) y � 1 = xyx � 1 < e , contradiction.

� A magma ( Q; � ) is a quandle if: 1. ( 8 a )[ a � a = a ] (idempotence); 2. for every b 2 Q , the mapping � b : Q ! Q de…ned by � b ( a ) = a � b is bijective; 3. ( 8 a; b; c )[( a � b ) � c = ( a � c ) � ( b � c )] (right self-distributivity). � A quandle Q is called trivial if the operation � is de…ned by ( 8 a; b )[ a � b = a ] : Every linear ordering of elements of Q is right invariant.

� For a group G , the conjugate quandle Conj( G ) is one with domain G and the operation � given by a � b = b � 1 ab . Then every bi-order on G induces a right order on Conj( G ) . Let P be a bi-order on G . Then ( 8 x; c )[( e; x ) 2 P ) ( e; c � 1 xc ) 2 P )] Using P , we de…ne R on Conj( G ) as ( 8 a; b )[( a; b ) 2 R , ( e; a � 1 b ) 2 P ] ; where e is the identity of G . The order R is right invariant because for ( a; b ) 2 R and c 2 Conj( G ) , ( e; ( a � c ) � 1 ( b � c )) = ( e; ( c � 1 a � 1 c )( c � 1 bc )) = ( e; c � 1 ( a � 1 b ) c ) 2 P . Since ( e; a � 1 b ) 2 P , we have ( a � c; b � c ) 2 R .

� Not all right orders on Conj( G ) are induced by bi-orders on G . It is possible to have BiO ( G ) = ; , while RO (Conj( G )) 6 = ; . Let G be an abelian group with torsion. Then BiO ( G ) = ; ; but Conj( G ) is a trivial quandle, so it admits many right orders. � n - quandle Q n : ( 8 a; b )[ b � a � n = b ] , where b � a � n = ( : : : ( b � a ) � a ) � � � � � a ) � a with n a ’s For n = 2 we have involutive quandle Q 2 : for every group de…ne b � a = ab � 1 a Then RO ( Q n ) = ; unless n = 1 .

� Topology de…ned on LO ( M ) by subbasis f S ( a;b ) g ( a;b ) 2 ( M � M ) � � where � = f ( a; a ) j a 2 M g : S ( a;b ) = f R 2 LO ( M ) j ( a; b ) 2 R g . � ( Dabkowska, Dabkowski, Harizanov, Przytycki, Veve, 2007 ) Let M be a magma with cardinality jMj = m � @ 0 . Then LO ( M ) is a compact space. By Vedenisso¤’s theorem, LO ( M ) can be homeomorphically embedded into the Cantor cube f 0 ; 1 g m . Moreover, LO ( M ) is a closed subspace of the Cantor cube f 0 ; 1 g m .

� If M is a countable magma, then LO ( M ) is metrizable. � If M = G is a group, we showed how we could also use Conrad’s theorem to establish that LO ( G ) is compact. � (Conrad, 1959) A partial left order P can be extended to a total left order on G i¤ for every f x 1 ; :::; x n g � G nf e g there are � 1 ; :::; � n , � i 2 f 1 ; � 1 g , such that 2 sgr (( P nf e g ) [ f x � 1 1 ; :::; x � n e = n g ) , where sgr ( A ) is the sub-semigroup of G generated by A .

� For a countable group G , LO ( G ) 6 = ; is homeomorphic to the Cantor set i¤ for any sequence ( a 0 ; b 0 ) ; :::; ( a k � 1 ; b k � 1 ) , S ( a 0 ;b 0 ) \ � � � \ S ( a k � 1 ;b k � 1 ) is either empty or in…nite. � (Sikora, 2004) The space LO ( Z n ) for n > 1 is homeomorphic to the Cantor set. (Dabkowska, 2006) The space LO ( Z ! ) is homeomorphic to the Cantor set. � (Linnell, 2006) The space of left orders of a countable left-orderable group is either …nite or contains a homeomorphic copy of the Cantor set. There are countable groups with in…nitely countably many bi-orders.

� ( Solomon, 1998 ) For every bi-orderable computable group G , there is a computable binary tree T and a Turing degree preserving bijection from BiO ( G ) to the set of all in…nite paths of T . � Hence, by the Low Basis Theorem of Jockusch and Soare, T has a low in…nite path. Recall that a set X and its Turing degree x are low if x 0 = 0 0 . Hence BiO ( G ) contains an order of low Turing degree. � (Metakides and Nerode, 1979) The sets of orders on computable …elds are in exact correspondence to the sets of � 0 1 classes.

� (Downey and Kurtz, 1986) There is a computable torsion-free abelian group with no computable order. � (Dobrica, 1983) Every computable torsion-free abelian group is isomorphic to a computable group with a computable basis. � Every computable torsion-free abelian group is isomorphic to a computable group with a computable order.

� (Harizanov, Knight, Lange, Puzarenko, Solomon, Wallbaum, 2011) Let be F 1 be the free group of rank @ 0 . (i) There is a computable copy of F 1 with no computable left order. (ii) Suppose F is a computable copy of F 1 , and let P be an order on F . Suppose B is a basis for F . Then for any X > T P � B , there is an order Q on F 1 such that Q � T X . (iii) There is a computable copy of F 1 with a computable order and no c.e. basis.

� Turing degree spectrum of left-orders on computable G : DgSp G ( LO ) = f deg( P ) j P 2 LO ( G ) g deg( P ) = deg( � P ) D = the set of all Turing degrees � (Solomon, 2002) (i) DgSp G ( LO ) = D for a torsion-free abelian group G of …nite rank n > 1 . (ii) DgSp G ( LO ) � f x 2 D j x � 0 0 g for a torsion-free abelian group G of in…nite rank. (iii) DgSp G ( LO ) � f x 2 D j x � 0 ( n ) g for a torsion-free properly n -step nilpotent group G .

� A group G for which every partial (left) order can be extended to a total (left) order is called fully orderable ( fully left-orderable ). Torsion-free abelian groups are fully orderable. � ( Dabkowska, Dabkowski, Harizanov, Togha, 2010 ) Let G be a computable, fully left-orderable group and d a Turing degree such that: ( a ) No left order on G is determined uniquely by any …nite subset of G nf e g ; ( b ) For a …nite A � G nf e g , the problem ‘ e 2 sgr ( A ) ’ is d -decidable; ( c ) DgSp G ( LO ) closed upward. Then DgSp G ( LO ) � f a 2 D j a � d g and LO ( G ) is homeomorphic to the Cantor set .

� Free group F n = h x 0 ; x 1 ; : : : ; x n � 1 j i of rank n > 1 is not fully left-orderable. � ( Dabkowska, Dabkowski, Harizanov, Togha, 2010 ) Let G be a computable group, d a Turing degree, P = f p i g i 2 ! a d -computable strong array of …nite subsets of G nf e g such that for every p 2 P , we have e = 2 sgr ( p ) and ( a ) there are a 2 G nf e g and q; r 2 P such that q � p ^ r � p and a 2 q ^ a � 1 2 r ; ( b ) for each a 2 G nf e g there is q 2 P such that q � p and a 2 q _ a � 1 2 q . Then ( 8 x � d )( 9 z 2 DgSp G ( LO ))[ x = z _ d ] :

� Corollary. If DgSp G ( LO ) is closed upward, then f x 2 D j x � d g � DgSp G ( LO ) . � ( Dabkowska, Dabkowski, Harizanov, Togha, 2010 ) For the free group F n of rank n > 1 , we have DgSp F n ( BiO ) = D . Proof idea : For a group G , the lower central series is the descending sequence of subgroups f � � ( G ) g de…ned as: � 0 ( G ) = G; � � +1 ( G ) = [ � � ( G ) ; G ] ; � � ( G ) = T �<� � � ( G ) , when � is a limit ordinal ; where [ A; B ] is the subgroup of G generated by the elements a � 1 b � 1 ab , with a 2 A and b 2 B .