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 : (8x; y; z)[x < y ) z x < z y] < is a bi-order (order) on the structure if (8x; 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 , y1 <l x1 G is left-orderable group ) G is torsion-free torsion-free: (8x 2 G feg)[order(x) = 1] e < x ) x < x2 < < xn (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 = fa 2 G j a eg Negative partial cone: P 1 = fa 2 G j a eg

• 1. PP P (P sub-semigroup of G)
• 2. P \ P 1 = feg (P pure)

P with 1 & 2 de…nes a partial left order P on G: x P y , x1y 2 P x P y ) x1y 2 P ) x1z1zy = (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. (8g 2 G)[g1Pg P] (P normal)

bi-order >: let g 2 G x > e ) g1xg > g1eg = e P normal: let x P y, z 2 G right invariant: x1y 2 P ) z1x1yz 2 P (xz)1yz 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 G =

D

x; y j xyx1y = e

E

left-orderable, but not bi-orderable. Positive cone P = fxnym 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 ) y1 = xyx1 > e y < e ) y1 = xyx1 < e, contradiction.

A magma (Q; ) is a quandle if:

• 1. (8a)[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. (8a; 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 (8a; 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 = b1ab. Then every bi-order on G induces a right order on Conj(G). Let P be a bi-order on G. Then (8x; c)[(e; x) 2 P ) (e; c1xc) 2 P)] Using P, we de…ne R on Conj(G) as (8a; b)[(a; b) 2 R , (e; a1b) 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; (c1a1c)(c1bc)) = (e; c1(a1b)c) 2 P. Since (e; a1b) 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 Qn: (8a; b)[b an = b], where b an = (: : : (b a) a) a) a with n a’s For n = 2 we have involutive quandle Q2: for every group de…ne b a = ab1a Then RO(Qn) = ; unless n = 1.

Topology de…ned on LO(M) by subbasis fS(a;b)g(a;b)2(MM) where = f(a; a) j a 2 Mg: S(a;b) = fR 2 LO(M) j (a; b) 2 Rg. (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 f0; 1gm. Moreover, LO(M) is a closed subspace of the Cantor cube f0; 1gm.

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 fx1; :::; xng Gnfeg there are 1; :::; n, i 2 f1; 1g, such that e = 2 sgr((Pnfeg) [ fx1

1 ; :::; xn 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 (a0; b0); :::; (ak1; bk1), S(a0;b0) \ \ S(ak1;bk1) is either empty or in…nite. (Sikora, 2004) The space LO(Zn) 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 x0 = 00. 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 F1 be the free group of rank @0. (i) There is a computable copy of F1 with no computable left order. (ii) Suppose F is a computable copy of F1, 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 F1 such that Q T X. (iii) There is a computable copy of F1 with a computable order and no c.e. basis.

Turing degree spectrum of left-orders on computable G : DgSpG(LO) = fdeg(P) j P 2 LO(G)g deg(P) = deg(P) D = the set of all Turing degrees (Solomon, 2002) (i) DgSpG(LO) = D for a torsion-free abelian group G of …nite rank n > 1. (ii) DgSpG(LO) fx 2 D j x 00g for a torsion-free abelian group G of in…nite rank. (iii) DgSpG(LO) fx 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 Gnfeg; (b) For a …nite A Gnfeg, the problem ‘e 2 sgr(A)’ is d-decidable; (c) DgSpG(LO) closed upward. Then DgSpG(LO) fa 2 D j a dg and LO(G) is homeomorphic to the Cantor set.

Free group Fn = hx0; x1; : : : ; xn1 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 = fpigi2! a d-computable strong array of …nite subsets of Gnfeg

such that for every p 2 P, we have e = 2 sgr(p) and (a) there are a 2 Gnfeg and q; r 2 P such that q p ^ r p and a 2 q ^ a1 2 r; (b) for each a 2 Gnfeg there is q 2 P such that q p and a 2 q _ a1 2 q. Then (8x d)(9z 2 DgSpG(LO))[x = z _ d]:

• Corollary. If DgSpG(LO) is closed upward, then

fx 2 D j x dg DgSpG(LO). (Dabkowska, Dabkowski, Harizanov, Togha, 2010) For the free group Fn of rank n > 1, we have DgSpFn(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 a1b1ab, with a 2 A and b 2 B.

Lower central series of Fn: 1(Fn) i(Fn) (Magnus)

1

\

i=1

i(Fn) = feg (Hall) i(Fn)=i+1(Fn) = Zki, where ki = 1

i

X

dji

( i

d)nd, Möbius function

Isomorphism uniformly computable since a basis of i(Fn)=i+1(Fn) can be found algorithmically in n; i.

Construct bi-orders on Fn using bi-orders on i(Fn)=i+1(Fn). Di¤erent choices of orders on quotients induce di¤erent orders on Fn: Produce a bi-order on Fn of a given Turing degree.

A set A is weak truth-table reducible to a set B: A wtt B if there is a computable function h and an index e so that A(x) = 'Bh(x)

e

(x): tt-reducibility, a stronger notion, is a further re…nement: A is truth-table reducible to B: A tt B if A wtt B via 'B

e and a computable function h,

having the additional robustness property: for any string 2 2<! of length h(x), '(x) #.

(Chubb, Dabkowski, Harizanov, 2011) Let G be a group, and P a c.e. family of …nite subsets of G feg satisfying the following conditions for every p 2 P: (purity) e 62 sgr(p); (branching) (9q; r 2 P)(9a 2 G)[q; r p ^ a 2 q ^ a1 2 r]; (extendability) (8a 2 G feg)(9q 2 P)[q p ^ (a 2 q _ a1 2 q)]. Then G admits a bi-order in each tt-degree.

We now generalize the construction for free groups of …nite rank > 1 to a class of …nitely presented, residually nilpotent groups that are not nilpotent. (Chubb, Dabkowski, Harizanov, 2011) Let G be a …nitely-presented, torsion-free, computable group. Let G = 1(G) 2(G) be the lower central series of G. If !(G) = feg and i(G)=i+1(G) is non-trivial and torsion-free for each i = 1; 2; : : : ; then there is a bi-order on G in every tt-degree.

Conjecture (Sikora, 2004) The space BiO(Fn) for n > 1 is homeomorphic to the Cantor set. (Navas-Flores, 2008) The space LO(Fn) for n > 1 is homeomorphic to the Cantor set.

THANK YOU!