On the free product of ordered groups

Transcription

1 rxiv: v [mth.gr] 6 Mr 207 On the free product of ordered groups A. A. Vinogrdov One of the fundmentl questions of the theory of ordered groups is wht bstrct groups re orderble. E. P. Shimbirev [2] showed tht free group on ny set of genertors cn be ordered. This leds to the following problem: under wht conditions is it possible to order free product of rbitrry groups? Using the mtrix presenttion method for groups proposed by Mlcev [], in the present work we estblish the orderbility of free product of rbitrry ordered groups. Definition. An ordered group is group endowed with reltion >, stisfying the following conditions:. For ny elements x nd y of the group either x > y, or y > x, or x = y. 2. If x > y nd y > z, then x > z. 3. If x > y, then xb > yb for ny elements nd b of the group. Definition 2. An ordered ring (field) is ring (field) such tht:. the dditive group of the ring (field) is ordered, nd 2. for ny elements, x, y of the ring (field), ( > 0 nd x > y) = (x > y nd x > y). Definition 3. The group lgebr kg of groupgover fieldkis the lgebr whose elements re forml finite liner combintions of elements of G with coefficients in k. These sums re multiplied nd dded in the usul wy. A group lgebr hs the obvious unit e, where e is the identity element of G nd the unit of k. Lemm. If k is n ordered field nd G n ordered group, then kg is orderble. Published in Mt. Sb. (N.S.), 949, Volume 25(67), Number, Trnslted from Russin by Victori Lebed nd Arnud Mortier.

2 Proof. Let A nd A be elements of kg under the conditions of the lemm. Then they cn be written s n n A = α i i, A = α i i, i= where some of the α i nd α i might be zero, nd >... > n. We set A > A if for some r {,..., n}, α = α,..., α r = α r, α r > α r. It is esy to check tht the conditions from Definition 2 hold. We cll tringulr mtrix ny mtrix, finite or infinite, with zeroes under the min digonl. Lemm 2. The set of ll tringulr mtrices with entries in n ordered unitl ring, nd with every element on the min digonl positive nd invertible, is n orderble group. Proof. Tringulr mtrices of the form described in the sttement clerly form group. Let X nd Y be such mtrices. We will cll preceding entries to given entry x ik, those x nm locted to the right of or on the min digonl, for which n m k i when m < i, nd n m < k i when m i. Sy tht X > Y if either of the following conditions holds: x ii = y ii for i =,..., k, nd x kk > y kk for some k, x ik > y ik for some k > i, nd their preceding entries coincide. One esily checks tht the conditions of Definition re stisfied. Lemm 3. The direct product of two ordered groups is orderble. Proof. Let A nd B be ordered groups. Sy tht (, b) > (, b ) in A B if either >, or = nd b > b. It is esy to check tht the conditions from Definition hold. We denote by M the direct product of two ordered groups A nd B. A pir of the form (, e ) where e is the identity of B will be denoted simply by, nd pir of the form (e, b) where e is the identity of A will be denoted by b. Trnsltors note: we believe tht there is mistke here, x nm should probbly be replced with x mn. 2 i=

3 Consider now the following trnscendentl tringulr mtrix: X = x 2 x 3 x 4 x 23 x 24 x 34 We denote by G the free belin group generted by the entries x ij of X. This group is orderble (see [2] nd references therein). By Lemm, the group lgebr K = QG is orderble, nd thus hs no zero divisors. The field of frctions Frc(K) of this lgebr is lso orderble [3]. Consider the group lgebr L = Frc(K)M, where M = A B s bove. According to Lemms nd 3, the lgebr L is orderble. Lemm 4. Consider the digonl mtrix A = where is the unit of L nd L is neither 0 nor. Then every entry of the mtrix B = X AX locted to the right of or on the min digonl is non-zero. Proof. Put X = (y ik ) nd B = (b ik ). Clerly 2, y in = x in + i<α <n x iα x α n i<α <α 2 <n +( ) n i x i, i+ x i+, i+2... x n, n x iα x α α 2 x α2 n Trnsltors note: we corrected the lst term of the formul given for y in. Note lso tht this formul holds only for i n, s y ii =. As result, the very lst formul of this proof is slightly incorrect when i is odd, but the min point tht the coefficient of b ik is not 0 seems to hold true fter ll. 3

4 nd b ik = (y i x k + y i3 x 3k + + y i, 2l+ x 2l+, k ) + (y i2 x 2k + y i4 x 4k + + y i, 2r x 2r, k ). From this follows: y i x k + y i3 x 3k + + y i, 2l+ x 2l+, k = x in x nk + i<α <n x iα x α nx nk i<α <α 2 <n x iα x α α 2 x α2 nx nk +, where the externl sums re over ll odd integers n between i nd k. This equlity shows tht the coefficient of in b ik is non-zero, nd so b ik 0. Theorem. The free product of two ordered groups cn be endowed with group order whose restriction to ech fctor is the originl order. Proof. Consider, together with the tringulr mtrix X introduced before, the following trnscendentl tringulr mtrices: Y = y 2 y 3 y 4 y 23 y 24 y 34, U = u u 2 u 3, V = v v 2 v 3. Let A nd B be ordered groups. As before, we construct n lgebr L = Frc(QG)M with M = A B, where now the free belin group G is generted by the set of ll forml entries not only of X, but lso of Y, U, 4

5 nd V. To every = (, e ) M we ssocite the digonl mtrix A =, nd to every b = (e, b) M the digonl mtrix b B b = b. Clerly the two sets of mtrices A = { } { } A A nd B = Bb b B form groups nturlly isomorphic to A nd B respectively. Put A = U X AXU nd B = V Y BY V. We re going to show tht the representtions of A nd B given by A nd b B b induce fithful representtion of the free product A B, tht is, given elements of A B of type ( n n ) n n r = i b i, r 2 = i b i k, r 3 = b k i b i, r 4 = b i i, the corresponding mtrices ( n n ) n n R = A i B i, R 2 = A i B i A k, R 3 = B k A i B i, R 4 = B i A i re not the identity mtrix. We will write down the proof for R only, s the three remining cses re similr. Every entry i kl of the mtrix A i is equl to u k kl iu l, where kl i is n entry of A i = X A i X, nd u k nd u l re digonl entries of the mtrices U nd U. Similrly, b i kl = vk b klv i l, where b kl i is n entry of B i = X B i X, nd vk nd v l re digonl entries of the mtrices V nd V. By Lemm 4, every mtrix in the groups A = X AX nd B = Y BY different from the identity mtrix hs only non-zero entries to the right of or 5

6 on the min digonl. The entries of the mtrix R re given by r ik = () ii 2 b () i 2 i 3 (2) i 3 i 4 b (2) i 4 i 5 (n) i 2n,i 2n b (n) i 2n,k. i i 2 i 3... i 2n k Here i k. This sum cn be regrded s polynomil in the digonl entries of U, V nd of their inverses. The coefficients of this polynomil re products of entries of the mtrices A, B, A 2, B 2,.... Observe tht no monomil occurs twice in the sum s it is given. Moreover, every coefficient is non-zero, since it is product of non-zero elements of the lgebr L, which hs no zero divisors. Therefore, we hve fithful representtion of the free product A B, given by r i R i. Every digonl entry of R i is either the unit of L or positive invertible element of L distinct from the unit. It follows then from Lemm 2 tht ll mtrices of ll four types R i together form n orderble group. Therefore, the free product A B is orderble. The proof presented here for two fctors obviously works for ny number of fctors. References [] A. Mlcev. On isomorphic mtrix representtions of infinite groups. Rec. Mth. [Mt. Sbornik] N.S., 8 (50): , 940. [2] H. Shimbirev. On the theory of prtilly ordered groups. Rec. Mth. [Mt. Sbornik] N.S., 20(62):45 78, 947. [3] B. L. vn der Werden. Modern Algebr. Vol. I. M. L.,