IBN AL- HAITHAM J. FOR PURE & APPL. S CI.          VOL.22 (4) 2009 

 
A Space of Fuzzy Orderings 

 
 

L.N. M. Tawfiq  
Departme nt of Mathematics, College of Education I bn-Al-Haitham , Unive rsity 
of Baghdad 
 

Abstract 
In this p aper the chain length of a sp ace of fuzzy  orderings is defined, and various 

p rop erties of this invariant are p roved. The st ructure theorem for sp aces of finite chain length 
is p roved. 

  

Spaces of Fuzzy Orderings 
        Throughout X = (X,A) denoted a sp ace of fuzzy  orderings. T hat is, A is a fuzzy 
subgroup  of abelian group  G of exp onent 2. (see [1] (i.e. x

2
 = 1,  x  G), and X is a (non 

emp ty) fuzzy subset of the character group   (A) = Hom(A,{1,–1}) satisfy ing: 

1. X is a fuzzy closed subset of  (A). 
2.  an element e  A such that (e) = – 1    X. 

3. X :={a  A\ (a) = 1    X} = 1. 
4. If f and g are for ms over A and if x  D( f  g) then  y  D( f ) and z  D(g) such that x 

 D<y, z >. 
        Observe, by  3, that the element e  A whose existence is asserted by 2 is unique. 
Also, e  1 (since (1) = 1    X). 
        Not ice that for a  A, t he set X(a):= {  X(a) = 1} is clopen (i.e. both closed and 
op en) in X. M oreover, (a) = – 1  (– a) = 1 holds for any    X     (by  2). 
 

Definition 1 
        A forms f and g are said to be isometric (over X) if they have the same dimension and 
(f ) = (g)    X. T his is denoted by  writing  f   g or                      g  f (over X). 
        Not e A form f is said to rep resent t he element X  A (over X) if  elements x1,…,xn  A 
such that f   < x, x2,…, xn >  D(f ) or D(f ,X) will be used to denote the set of elements of A 
which are rep resented by  f  in this sense. 
 

Definition 2 
   A form f is said to be isotrop ic if  x3, …, xn  A, such that                           
  f   <1,–1, x3, …, xn >. Not ice, in p articular, this imp lies dim(f )  2. A form which is not 
isotrop ic is said to be anisot rop ic, for any x  A, < x,– x >  <1,–1>. 
any such form will be called a hy p erbolic p lane. 
 
The orem 1 
        The following are equivalent 
(i)  x  G, x  – 1  D<1, x > = {1, x}. 
(ii)  X = {   (A)(–1) = –1}. 
Proof: see [3]. 
        A sp ace of fuzzy  ordering satisfy ing either of the equivalent conditions in theorem 1 will 
be referred to as a fan. 
 



IBN AL- HAITHAM J. FOR PURE & APPL. S CI.          VOL.22 (4) 2009 
 

Corollary 1 
        Sup p ose X is a fan. Then every  subsp ace of X is also a fan. 
Proof: comp are [3]. 
        Recall, a sp ace of fuzzy  orderings (X,A) is said to be finite if X   (or equivalently A) is 
finite fuzzy  set; and two sp aces of fuzzy  orderings (X,A) and (X,A) are said to be 
isomorphic if there exists a group  isomorphism :A  A such that t he dual isomorp hism 
*:(A)  (A) maps X on to X. 
 

Definition 3 
        The chain length of X (denoted C1(X)) is the maximum integer k  1 such that  a0, , 
ak  A satisfy ing: X(ai – 1)  X(ai ), i = 1, , k (or C1(X) =  if no such maximum exists). 
 
Remark 1 
        It is easily verified that C1(X) = 1 if and only  if x = 1, and C1(x)  2 if and only if X is 
a fan. 
        Recall that X is said to be decomp osable if there exist non-emp ty  subsp aces Xi of X, i = 
1,2 such that X = X1  X2. 
Let us denote by gr(X) t he translation fuzzy  group  of X, i.e., 
gr(X) = {T   (x)  TX = X}. 
Thus gr(X) is a closed fuzzy  subgroup  of (A). 

        Let the residue sp ace of X be defined to be X = (X,A) where A=gr(X)

A, and 

where X denotes the image of X in (A) via restriction, X is a sp ace of fuzzy  orderings. 
M oreover gr(X) = 1, and X is a fuzzy  group  extension of X. 
          We can st ate the main theorem concerning sp aces of finite chain length. 
 
The orem 2 
        Sup p ose C1(X) < . Then either X = 1, or gr(X)  1, or X is decomp osable. 
        The p roof of this key result is found in [4]. For now we concentrate on giving two 
imp ortant ap p lications. 
 
The orem 3 
        Sup p ose a form f  is anisot rop ic over a sp ace of fuzzy  ordering X0. Then there exists a 
finite subsp ace X  X0 such that f is an isot rop ic over X. 
Proof:  Let X=(X,A) be a subsp ace of X0 chosen minimal subject to f is anisot rop ic over X. 
Let a0, , ak  A satisfy : D<1,ai – 1>  D<1,ai >, i = 1,,k. Thus <1,ai >  < ai – 1, ai – 1 ai > 
and ai – 1  ai  for i = 1,,k. We may  assume a0 = 1, ak = 1. Let bi = ai – 1 ai . Thus bi  1, so 
X(bi) is a p rop er subsp ace of X. Thus f is isotrop ic over X(bi), i.e. there exists a form gi of 
dimension n – 2 (where n denotes t he dimension of f) such that f  gi over X(bi). Thus: 
f  <1,bi >  gi  <1,bi > over X, so by  addition 
 

k k

i 1 i 1

1, 1,
i i i

f b g b
 

       
 
 :  (over X) (1) 

 
But using the assump tions on a0, , ak we see that (over X) < b0, , bk >               < a0 a1, 
a1 a2,, ak – 1 ak >  < a1, a1 a2,, ak – 1 ak >  < 1, a2, a2 a3,, ak – 1 ak >  < 1,, 1, ak 
>  < 1,,1, 1 >. 
Subst itut ing this in (1) yields 

k

i 1

(2  - 2) 1,
i i

k  f g b


 :  



IBN AL- HAITHAM J. FOR PURE & APPL. S CI.          VOL.22 (4) 2009 
 

Now f (and hence (2k – 2) f, by  (3, corollary  3.5(ii)) is anisot rop ic over X, so comp aring 

dimensions, and using (3, lemma 2.4), (2k – 2) n  k (n – 2)(2), i.e.,           k  
1

2
n. This 

p roves C1(X) < . 
Now, we app ly theorem 2. If X = 1 we are done. 
Sup p ose X = X1  X2 where Xi = (Xi,A /i) is a non-empty  subsp ace of X, i = 1,2. Thus t here 
exist elements ai 3, , ai n  A such that 
f  <–1,1, ai 3, , ai n > over Xi, i = 1,2. 
Since X = X1  X2, the natural injection A  A /1  A /2 is surjective, so t here exist a3, 
, an  A such that aj  aij (mod i), 3  j  n, i = 1,2. 
Then clearly f  <1,–1, a3,,an > over X, a contradiction. Thus X is indecomp osable, so 
gr(X)  1. Let X=(X,A) denote the residue sp ace of X and decomp ose f as f 1 f1  s 
fs where f1,,fs are forms over A, and 1,,sA are dist inct modulo A. 
The assertion that f is anisot rop ic over X is equivalent to the assertion that each f1,,fs is 
anisot rop ic over X. 
There are two cases to be considered. 
        Sup p ose S = 1. Let  be any  fuzzy  subgroup  of A such that A is t he direct p roduct A =  

 A, and let Y = 


  X. Then one verifies easily that Y = (Y,A/) is a subsp ace of X and 
that (Y,A/)  (X,A), this equivalence being induced by  the natural isomorphism A/  A  
Thus, since f1 is anisot rop ic over X, it (and then f  1 f1) is anisot rop ic over Y. But, on the 
other hand gr(X)  1, i.e. A  A, i.e.   1, i.e., Y  X. 
This contradicts t he minimal choice of X. 
        Thus S  2. It follows that each fi has st rictly lower dimension than f so by  induction on 
the dimension, there exist finite subsp aces Z1, , Zs   X such that fi is anisot rop ic over Z i  
. Thus f1,,fs are all anisotrop ic over the subsp ace of X generated by  Z1, , Zs . Denote this 

sp ace by  Z = (Z,A/). Not e Z is st ill finite Z = 


  X. T hen Z = (Z,A/) is a subsp ace 
of X, and a fuzzy  group  extension of Z = (Z,A/). M oreover, since 1,,s are distinct 
modulo A, f is anisot rop ic over Z. Thus, by  minimal choice of X, Z = X, i.e.  = 1, i.e., Z = 
X is finite. However, X itself could be infinite (since, a priori, gr(x) could be infinite). Define 
A to be the fuzzy  subgroup  of A generated by  A and 1,,s, and let X denote the 
restriction of X to A. 
        Thus (X,A) is a fuzzy  group  extension see[ 2] of (X,A) which, inturn, is a fuzz y  group 
extension of (X,A). M oreover (X,A) is finite, and f is anisot rop ic over X. Finally, let  be 

fuzzy  subgroup  of A so that A =A, and let Y=

X. T hen Y = (Y,A/) is a subsp ace of 

X naturally  equivalent t o (X,A). Thus Y is finite, and f is anisot rop ic over Y. Thus Y = X is 
finite. 
        Not ice, the condition X(ai – 1)  X(ai) is equivalent t o D<1,ai >  D<1,ai – 1>. 
 
The orem 4 
(i) Sup p ose Xi = (Xi,A/i), i =1,,n are subsp aces of X generating X. Then:        CL(X) 

=
n

i 1

CL(X )
i


 . 

(ii)  If, in addition, X = X1  Xn, then: C1(X) = 
n

i 1

C1(X )
i


 . 

(iii) If X is a fuzzy  group  extension of X, then CL(X) = CL(X), excep t in the case X = 1 
(in which case X is a fan). 

 
 



IBN AL- HAITHAM J. FOR PURE & APPL. S CI.          VOL.22 (4) 2009 
 

Proof: 
(i) Sup p ose X(aj – 1)  X(aj), j = 1,,k. Then for each i, 1  i  n, Xi(aj – 1)  Xi(aj). 

M oreover, since X(aj – 1)  X(aj), there exists i, 1  i  n such that  Xi(aj – 1)  Xi(aj). (for 

if Xi(aj – 1) = Xi(aj) for all i  n, then aj aj – 1 
n

i 1
1

i

  , i.e., aj = aj – 1 a contradiction). 

This holds for j = 1,,k. Simp le counting y ields 
n n

i 1 i 1

k CL(X ), i.e., CL(X) CL( X )
i i

 

   . 

(ii) We are assuming X = Ui Xi and the natural homomorp hism from A into   iA /i is an 
isomorphism. Sup p ose Xi(ai,j – 1)  Xi (ai,j), j = 1,,ki, i = 1,,n. 

We may  as well assume ai,0 = – 1, and a 1
ii,k
 . Choose elements bij  A such that: bij 

= 1 (mod k) for k < i. 
bij  aij  (mod i), and bij  – 1 (mod k), for k > i. 

Not ice that X(bij) = (Us < i Xs) U Xi(aij). It follows that X(b10)  
11

X( )kb  = X(b20) 

 
n

X ( )n kb . There are  ki inequalities in this chain, so CL(X)    ki , and hence 

CL(X)   CL(Xi). 
The other inequality  follows from (i). 

(iii) Sup p ose X 1. Supp ose X(ai – 1)  X(ai), i = 1,,k, with ai  A. Then clearly   
X(ai – 1)  X(ai), i = 1,,k. Thus CL(X)  CL(X). Now supp ose D<1,ai >  D<1, 
ai – 1>, i = 1,,k, with a1, , ak  A. We may assume a0 = – 1, ak = 1. Then a1  – 1. 
There are two cases to be considered 
1

st
 Case: Sup p ose a1  A. It follows (from the definition of fuzzy  group  extension) that 

D<1,a1> = {1,a1}. T hus K  2 in this case. Thus, since    X 1, CL(X)  2  k. 
2

nd
 Case: Sup p ose a1  A. Then D<1,a1>  A (e.g. by  (5, lemma 4.9); notice a1  – 1. 

Thus a1, , ak are all in A, and X(ai – 1)  X(ai), i = 1,,k. Thus CL(X)  K. T hus, 
in any  case CL(X)  K, so CL(X)  CL(X). 
 

Lemma 1 
        Sup p ose b, a0, , ak  A satisfy  D<1,b> = {1,b}, and D<1,ai – 1> <1,b>  D<1,ai > 

<1,b>, i = 1,,k. Then there exists ia   D< ai,aib > = {ai,aib} such that D<1, i 1a   >  

D<1, ia >, i = 1,,k. 
Proof:  comp are [6]. 
        We now p roceed to p rove a deep er prop erty  of chain length. 
 
The orem 5 
        Sup p ose Y is a subsp ace of X. Then C1(Y)  C1(X). 
Proof:  Sup p ose, to the contrary , cl(Y) > cl(X). Then, in p articular, cl(X) < . Choose a 
subsp ace Z  X minimal subject t o (1)Z  Y and (2) C1(Z)  C1(X). T o show such Z exists. 
Sup p ose {Z i} is a collection of subsp aces of X satisfy ing (1) and (2) and linearly ordered by  

inclusion. Let z = 
i
Zi. Then z is a subsp ace of X satisfy ing (1). To show z satisfies (2) 

sup p ose a0, , ak A satisfy  z(aj)z(a j – 1), j = 1,,k. Thus t he set M ={X < 1, aj   
< a j – 1, a j – 1 aj >, j = 1,,k} is op en in X and contains Z. By  comp actness, Z i  M  for some 
i, so Zi(aj) Zi(a j – 1), j = 1,,k. These inclusions must be st rict, since Z  Zi. Thus k  
CL(Zi)  CL(X), so CL(Z)  CL(X). So Z exists as asserted. To simplify  notation, we may  
assume X = Z. Let Y = (Y,A/), since Y  X(CL(Y) > CL(X)). It  follows that   1, so t here 
exists a  , a  1. Thus Y  X(a)  X. Since CL(X) < , there exists b A, b  1, such that 



IBN AL- HAITHAM J. FOR PURE & APPL. S CI.          VOL.22 (4) 2009 
 

 X(a)  X(b)  X, X(b) maximal. Thus D<1,b> is minimal, i.e., D<1,b> = {1,b}. By  the 
minimal choice of X (=Z ), it follows that CL(X(b)) > CL(X). On the other hand it follows 
from lemma (1) that CL(X(b))  CL(X). This is a contradiction. 
 

Re ferences 
1. M alik ,D.S. and M ordeson, J.N. (1991),Fuz zy  subgroup s of Abelian group , Chinese 

J.M ath., 19(2). 
2. M ordeson ,J.N. and Sen,M .K. (1995), Basic Fuzzy  subgroup s, Inform Sci., 82, 167-179. 
3. M arshall ,M . (1980), The Witt ring of a sp ace of ordeeerings,T rans. Amer. M ath. Soc.258. 
4. M arshall, M . (1989), Ouot ients and inverse limits of sp aces of orderings, Can. J.M ath. 

31,604-616. 
5. M arshall,M . (1989), Classification of finite sp ace of orderings,Can. J.M ath. 31, 320-330. 
6. M arshall,M . (1990), Sp aces of orde ngs IV,Can. J.M ath., XXXII(3): 603-627. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 



 
(22مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة                 المجلد  4 (2009 

 
 

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