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

 
A Space of Fuzzy Orderings 

 
 

L.N.M.Tawfiq  
Department of Mathematics, College of Education Ibn-Al-Haitham , University 
of Baghdad 
 

Abstract 
In this paper the chain length of a space of fuzzy orderings is defined, and various 

properties of this invariant are proved. The structure theorem for spaces of finite chain length 
is proved. 

  

Spaces of Fuzzy Orderings 
        Throughout X = (X,A) denoted a space of fuzzy orderings. That is, A is a fuzzy 
subgroup of abelian group G of exponent 2. (see [1] (i.e. x2 = 1,  x  G), and X is a (non 

empty) fuzzy subset of the character group  (A) = Hom(A,{1,–1}) satisfying: 

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 forms 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). 
        Notice that for a  A, the set X(a):= {  X(a) = 1} is clopen (i.e. both closed and 
open) in X. Moreover, (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. This is denoted by writing  f   g or                      g  f (over X). 
        Note A form f is said to represent the 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 represented by f  in this sense. 
 

Definition 2 
   A form f is said to be isotropic if  x3, …, xn  A, such that                           
  f   <1,–1, x3, …, xn >. Notice, in particular, this implies dim(f )  2. A form which is not 
isotropic is said to be anisotropic, for any x  A, < x,– x >  <1,–1>. 
any such form will be called a hyperbolic plane. 
 
Theorem 1 
        The following are equivalent 
(i)  x  G, x  – 1  D<1, x > = {1, x}. 
(ii)  X = {   (A)(–1) = –1}. 
Proof: see [3]. 
        A space of fuzzy ordering satisfying either of the equivalent conditions in theorem 1 will 
be referred to as a fan. 
 



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

Corollary 1 
        Suppose X is a fan. Then every subspace of X is also a fan. 
Proof: compare [3]. 
        Recall, a space of fuzzy orderings (X,A) is said to be finite if X   (or equivalently A) is 
finite fuzzy set; and two spaces of fuzzy orderings (X,A) and (X,A) are said to be 
isomorphic if there exists a group isomorphism :A  A such that the dual isomorphism 
*:(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 decomposable if there exist non-empty subspaces Xi of X, i = 
1,2 such that X = X1  X2. 
Let us denote by gr(X) the 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 space 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 space of fuzzy orderings. 
Moreover gr(X) = 1, and X is a fuzzy group extension of X. 
          We can state the main theorem concerning spaces of finite chain length. 
 
Theorem 2 
        Suppose C1(X) < . Then either X = 1, or gr(X)  1, or X is decomposable. 
        The proof of this key result is found in [4]. For now we concentrate on giving two 
important applications. 
 
Theorem 3 
        Suppose a form f  is anisotropic over a space of fuzzy ordering X0. Then there exists a 
finite subspace X  X0 such that f is an isotropic over X. 
Proof:  Let X=(X,A) be a subspace of X0 chosen minimal subject to f is anisotropic 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 proper subspace of X. Thus f is isotropic over X(bi), i.e. there exists a form gi of 
dimension n – 2 (where n denotes the 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 assumptions 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 >. 
Substituting this in (1) yields 

k

i 1

(2  - 2) 1,
i i

k  f g b


 :  



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

Now f (and hence (2k – 2) f, by (3, corollary 3.5(ii)) is anisotropic over X, so comparing 

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

2
n. This 

proves C1(X) < . 
Now, we apply theorem 2. If X = 1 we are done. 
Suppose X = X1  X2 where Xi = (Xi,A /i) is a non-empty subspace of X, i = 1,2. Thus there 
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 there 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 indecomposable, so 
gr(X)  1. Let X=(X,A) denote the residue space of X and decompose f as f 1 f1  s 

fs where f1,,fs are forms over A, and 1,,sA are distinct modulo A. 
The assertion that f is anisotropic over X is equivalent to the assertion that each f1,,fs is 
anisotropic over X. 
There are two cases to be considered. 
        Suppose S = 1. Let  be any fuzzy subgroup of A such that A is the direct product A =  

 A, and let Y = 


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

space by Z = (Z ,A/). Note Z is still finite Z = 


  X. Then Z = (Z,A/) is a subspace 
of X, and a fuzzy group extension of Z = (Z,A/). Moreover, since 1,,s are distinct 
modulo A, f is anisotropic 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 fuzzy group 
extension of (X,A). Moreover (X ,A) is finite, and f is anisotropic over X . Finally, let  be 

fuzzy subgroup of A so that A =A , and let Y=

X. Then Y = (Y,A/) is a subspace of 

X naturally equivalent to (X ,A). Thus Y is finite, and f is anisotropic over Y. Thus Y = X is 
finite. 
        Notice, the condition X(ai – 1)  X(ai) is equivalent to D<1,ai >  D<1,ai – 1>. 
 
Theorem 4 
(i) Suppose Xi = (Xi,A/i), i =1,,n are subspaces 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), except in the case X = 1 
(in which case X is a fan). 

 
 



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

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

Moreover, since X(aj – 1)  X(aj), there exists i, 1  i  n such that  X i(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. Simple counting yields 
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 homomorphism from A into   iA /i is an 
isomorphism. Suppose 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. 

Notice that X(bij) = (Us < i Xs) U X i(aij). It follows that X(b10)  
11X( )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) Suppose X 1. Suppose 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 suppose 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: Suppose a1  A. It follows (from the definition of fuzzy group extension) that 
D<1,a1> = {1,a1}. Thus K  2 in this case. Thus, since    X 1, CL(X)  2  k. 
2nd Case: Suppose 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. Thus, 
in any case CL(X)  K, so CL(X)  CL(X). 
 

Lemma 1 
        Suppose 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:  compare [6]. 
        We now proceed to prove a deeper property  of chain length. 
 
Theorem 5 
        Suppose Y is a subspace of X. Then C1(Y)  C1(X). 
Proof:  Suppose, to the contrary, cl(Y) > cl(X). Then, in particular, cl(X) < . Choose a 
subspace Z  X minimal subject to (1)Z  Y and (2) C1(Z)  C1(X). To show such Z exists. 
Suppose {Z i} is a collection of subspaces of X satisfying (1) and (2) and linearly ordered by 

inclusion. Let z = 
i
Z i. Then z is a subspace of X satisfying (1). To show z satisfies (2) 

suppose a0, , ak A satisfy  z(aj)z(a j – 1), j = 1,,k. Thus the set M ={X < 1, aj   
< a j – 1, a j – 1 aj >, j = 1,,k} is open in X and contains Z. By compactness, Z i  M for some 
i, so Zi(aj) Z i(a j – 1), j = 1,,k. These inclusions must be strict, 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 there 
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. SCI.          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. 
 

References 
1. Malik ,D.S. and Mordeson, J.N. (1991),Fuzzy subgroups of Abelian group, Chinese 

J.M ath., 19(2). 
2. Mordeson ,J.N. and Sen,M.K. (1995), Basic Fuzzy subgroups, Inform Sci., 82, 167-179. 
3. Marshall ,M. (1980), The Wittring of a space of ordeeerings,Trans. Amer. Math. Soc.258. 
4. Marshall, M. (1989), Ouotients and inverse limits of spaces of orderings, Can. J.M ath. 

31,604-616. 
5. Marshall,M. (1989), Classification of finite space of orderings,Can. J.M ath. 31, 320-330. 
6. Marshall,M. (1990), Spaces of orde ngs IV,Can. J.M ath., XXXII(3): 603-627. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 



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

 
 

 الفضاء الضبابي الترتیب

 
 
 
 

  لمى ناجي محمد توفیق

  جامعة بغداد، ابن الهیثم -كلیة التربیة ،قسم الریاضیات 

 
 

  خالصةال

        یعـرض البحـث تعریــف طـول سلــسلة فـي فـضاء ضــبابي الترتیـب ومــن ثـم عـرض خــواص وبرهنتهـا، ولقــد تـم برهــان 

  .المبرهنة األساسیة لطول السلسلة المنتهیة وعرض بعض النتائج المتعلقة بالموضوع