CONSTRUCTION OF FINITE CYCLIC GROUPS WITH FUZZY
EQUALITIES.
M. Bradic
Ph.D student, University of Novi Sad, Novi Sad, Serbia
In this paper a characterization of finite cyclic L-groups is given, where L is a standard residuated lattice.
That characterization is useful to find a procedure for constructing all finite cyclic L-groups with a given
skeleton. As an example, for a given cyclic group G of order 8 [resp. 9] an algorithm for generating all Lgroups with skeleton G will be described.
Introduction
Fuzzy approaches to various universal concept started with Rosenfeld’s paper [1].
Algebras with fuzzy equalities (or L-algebras, where L is a residuated lattice) have two
parts: ordinary algebras (so called skeletons) and “relational part”, and they are structures
for the equational fragment of fuzzy logic. A more general structure can be found in [2].
When L is the two-element Boolean algebra, then ordinary algebras can be interpreted as
L-algebras, so such approach generalizes various results of universal algebras (for
example, see [3], [4], [5]). Since the carrier set of an L-algebra is equipped with a fuzzy
equality which is compatible with all of the fundamental operation of the ordinary part,
the theory of algebras with fuzzy equalities deals with compatible fuzzy equalities. Some
questions concerning fuzzy operations compatible with a fuzzy equivalence relation on
groups in a different framework can be found in [6]. In that paper, among other things,
were investigated so called T-vague groups of a group (T is a given t-norm). That other
direction of studying compatible fuzzy equalities was developed in [7]. A general
panorama of fuzzy equalities is presented in [8]. Our subject of study in this paper is
finite cyclic groups with fuzzy equalities (or finite cyclic L-groups), and the main problem
we are interested in is how to construct finite cyclic L-groups with the given skeleton.
Preliminaries
In this paper, residuated lattices will be used, as structures of truth values. A residuated
lattice is an algebra L  L, , , , , 0,1 such that L, , , 0,1 is a lattice with the
least element 0 and the greatest element 1, L, ,1 is a commutative monoid and  is a
binary relation, where  and  satisfy the adjointness property, i.e. for all x, y, z  L,
x  y  z if and only if x  y  z. If L, , , 0,1 is a complete lattice then L is said to
be complete. The operation  is called residuum, and the operation  in L defined by
( x  y )  ( x  y )  ( y  z ) is called biresiduum. It is easy to see that each residuated
lattice satisfies the following conditions: x  y iff x  y  1, x  y  1 iff x  y.
x  y  x, x  y  1 iff if x  y  1. . We will write x  2 instead of x  x.
Each of the following three pairs of adjoint operations makes a complete residuated
lattice [0,1], min, max, , , 0,1 ( [0,1] is the real unit interval):
a  b  max (a  b  1, 0), a  b  min (1  a  b,1) - standard Lukasiewicz structure;
b
a  b  a  b, a  b  1 if a  b, a  b 
otherwise – standard product structure;
a
a  b  min (a, b), a  b  1 if a  b, a  b  b otherwise – standard Gödel structure.
A binary fuzzy relation (or, L-relation) on X is any mapping R : X 2  L, and a operation
X n  X is compatible with R if for any u1 ,, u n , v1 ,, vn  X it holds:
R(u1 , v1 )    R(u n , vn )  R( f (u1 ,, u n ), f (v1 ,, vn )). A binary L-relation E on X is a
L-equivalence if E is reflexive: E ( x, x)  1, for all x  X , symmetric: E ( x, y )  E ( y, x),
for all x, y  X , and transitive: E ( x, y )  E ( y, z )  E ( x, z ), for all x, y, z  X . An Lequivalence is an L-equality if E ( x, y )  1 implies x  y , for all x, y  X .
An algebra with L-equality (shortly, L-algebra) of type F is a triplet M  M ,  M , F M ,
where M  M , F M
is an ordinary algebra of type F and  M is an L-equality on M
such that each fundamental operation of M is compatible with  M . The ordinary part M
is called skeleton of M . An L-algebra is said to be L-group (or, a group with fuzzy
equalities) if its skeleton is an ordinary group.
Finite cyclic L-groups
In the sequel, L  L, , , , , 0,1 will be an arbitrary complete residuated lattice.
An L-group G,  G , , 1, e is said to be a finite cyclic L-group if its skeleton is a finite
cyclic group.
Lemma 1. Let G  G, , 1, e be a finite cyclic group generated by a, and let  be a
binary L-relation on G. Then, the structure G  G, ,  , 1, e is a cyclic L-group if and
only if  satisfies the following three conditions, for all i, j  Z :
(i)
(e a i )  1 if and only if a i  e;
(ii)
(a i a j )  (e a i  j );
(iii)
(e a i )  (e a j )  (e a i  j ).
The lemma above gives a characterization of all finite cyclic L-groups. Note that it is only
a direct consequence of a general result concerning arbitrary finite L-groups, which was
established in [9], where also the notion of finite cyclic L-spectrum was introduced, by
the following definition:
Definition 2. Let 0  n  N and m  [n / 2] (the integer part of n / 2 ). A sequence
s0 , s1 ,, sm of elements from L is said to be a cyclic L-spectrum of order n if s0  1,
si  1, for all i  0, and the following inequalities hold, for all i, j, 0  i, j  m :
if i  j  m
 si  j ,
si  s j  
;
s n(i  j ) , if i  j  m
si  j , if i  j
si  s j  
.
s j i , if i  j
Theorem 3. Let G  G, , 1, e be a cyclic group of order n generated by a, m  [n / 2] ,
and let   s0 , s1 ,, sm be a cyclic L-spectrum of order n. Let  be L-relation on G
defined by (e   a i )  si , for all i, 0  i  m, and (a i   a i )  (e   a i  j ) in all the
remaining cases. Then, the structure G   G,   ,  , 1, e is an L-group.
Furthermore, there is an one-to-one correspondence between the set of all L-groups with
skeleton G and the set of all L-spectra of order n.
The theorem above was previously established in [9]. A consequence of that theorem is
that a characterization of the set of all finite cyclic L-spectra of order n implies a
characterization of all L-groups with skeleton G, where G is a cyclic group of order n.
A characterization of finite cyclic L-spectra
Before preceding, note that each residuated lattice L satisfies the following condition:
x  ( y  z ) if and only if x  y  z and x  z  y.
(1)
From now, L is assumed to be a standard residuated lattice.
Therefore, we have that in L for each positive a there is a unique b such that b 2  a , and
now for each standard structure we will define an unary operation in the following way:
Definition 4. For all a  [0,1], we let the following:
a 1

a :
, if L is the standard Lukasiewicz structure;
(i)
2

a : a , if L is the standard product structure;
(ii)
(iii)

..
a : a, if L is the standard G o del structure.
It can easily be shown that L satisfies the following condition:
x 2  y if and only if x   y .
(2)
Definition 5. Let 4  n  N and m  [n / 2]. Let s0 , s1 ,, sm be a sequence of elements
from [0,1]. For each 2  k  m, we set:
 k  si  s j | 1  i  m, 1  j  m, i  j  k , i  j;
 k  si  sn(i k ) | 1  i  k , 1  n  (i  k )  k ;
 k   si  s j | 1  i  m, 1  j  m, i  j  k , i  j;


 k   si  sn(i k ) | 1  i  k ,1  n  (i  k )  k .
The main result in this paper is the following proposition..
Proposition 6. Let 4  n  N and m  [n / 2] . Let
s0 , s1 ,, sm
be a sequence of
elements from [0,1]. Then, s0 , s1 ,, sm is a cyclic L-spectrum of order n if and only if
the following conditions hold:
(i)
s0  1, and si  1, for all i, 1  i  m;
n
(ii)
 k   k  sk   k   k , for all k, 2  k  ;
3
n
 k   k  sk   k   k   s n2k , for all k,  k  m.
(iii)
3
Proof. Since s0  1 and  is commutative, it is easy to verify that s0 , s1 ,, sm is a
cyclic L-spectrum of order n if and only if the following conditions hold:
(a) s0  1, and si  1, for all i, 1  i  m;
(b) si  s j  si  j , for all i, j, 1  i, j  m such that i  j  m and i  j;
(c) si  s j  s n(i  j ) , for all i, j, 1  i, j  m such that m  i  j  n;
(d) si  s j  si  j , for all i, j, 1  i, j  m such that j  i.
Now, suppose that s0 , s1 ,, sm satisfies the conditions (a)-(d). We will prove that
under this assumption the conditions (i)-(iii) hold. (i) is trivially true. Take any k, where
2  k  m. Note that hence m  n  k. Suppose that k  i  j. Then, by (1) and (d) we
obtain si  j  ( si  s j ), and if i  j , then si  s j  si  j , by (b). Thus, we can obtain
 k  sk   k . Now, suppose that there is i such that 1  i, n  (i  k )  k . Hence,
m  n  i and m  i  k  n, so s k  s i  s n (i  k ) and s k  s n (i  k )  s n ( ni )  si , by (c).
These inequalities by (1) imply s k  ( si  s n(i  k ) ). So, we can get sk   k . Further,
since m  n  k. , if m  n  k then s i  s n (i  k )  s n ( n  k )  s k , by (c), and if m  n  k
then n  2m and k  m, so s i  s n (i  k )  s k means si  smi  sm , which is true by (b).
Thus, we can get  k  sk . Finally, suppose that k  n / 3. Then m  2k , so by (c) we
have s k 2  s n2k , hence by (2) we obtain s k   s n2 k , as desired.
s0 , s1 ,, sm satisfies the conditions (i)-(iii). We need to
prove that under this assumption the conditions (a)-(b) hold. Let 1  i, j  m . Trivially,
Conversely, suppose that
(a) holds. To prove (b) we suppose that i  j  m and i  j. Now,  i  j  si  j implies
(b). To prove (d) we suppose that j  i. Then, using si   i we obtain si  ( s j  si  j ),
hence by adjointness property si  s j  si  j . Thus, (d) holds. Further, to prove (c) we
suppose that m  i  j  n. We want to prove si  s j  s n (i  j ) . The case n  (i  j )  i
is trivial, as well as the case n  (i  j )  j. If i  j , n  (i  j ), then si   i implies
si  ( s j  s n (i  j ) ), hence the result follows by adjointness property. The case
j  i, n  (i  j ). is analogous to the previous one. If n  (i  j )  i, j , then we put
p  n  (i  j ), hence p  2 and n  (i  p )  j , and now from  p  s p we obtain
s i  s n (i  p )  s p , i.e. si  s j  s n (i  j ) . It remains the case i  j  n  (i  j ), where we
need to prove si 2  s n2i . So, let i  n  2i. Since i  n / 3, we have si   s n2i , from
(iii). By (2), it means si 2  sn2i , as desired. Thus, (c) holds, which finishes the proof of
proposition.
Thus, the set of all cyclic L-spectra of order n, m  [n / 2], can be characterized by a
system of inequalities: x0  1, x1 2  x2  1,  3  x3   3  1, … ,  m  xm   m  1,
such that for all i, k, 3  k  m, k  i  m, no xi occurs in formulas  k and  k . That
system is not “solved” because in general any choice of values for x1 and x 2 satisfying
x1 2  x2  1 does not imply that all the remaining intervals  k ,  k  are correct. It is
necessary to find a formula  2   2 ( x1 ) such that x2   2 . But, the previous result can
be useful to some particular cases, as will be seen in the following corollary.
Corollary 7. For a given cyclic group G of order 8 [resp. 9] there is an algorithm for
obtaining all L-groups with skeleton G.
Proof. By Proposition 6, a sequence s0 , s1 , s2 , s3 , s4 is a cyclic L-spectrum of order 8 if
and only if it is a solution in L of the following system of inequalities:
x0  1, x1  1,
x1 2  x2  1,
x1  x2  x3  ( x1  x2 )   x2 ,
x4  1, ( x1  x3 )  x2 2  x4  x1  x3 .
It is easy to see that this system is “solved” (L satisfies (1) and x1  x2  x2   x2 , and
in L, x2 2  x1  x3 follows from x1  x2  x3 and x1  x3  x1 ). So, we can describe
an algorithm for obtaining all cyclic L-spectra of order 8. Then, by Theorem 3, it is
obvious that there is an algorithm for constructing all L-groups with skeleton G, where G
is a cyclic group of order 8. Thus, the first part of the corollary is proved.
Also, by Proposition 6, a sequence s0 , s1 , s2 , s3 , s4 is a cyclic L-spectrum of order 9 if
and only if it is a solution in L of the following system of inequalities:
x0  1, x1  1,
x1 2  x2  1,
x3  1, x1  x2  x3  x1  x2 ,
( x1  x3 )  x2 2  x4  ( x1  x3 )   x1 .
It is needed to find some “solved” system of inequalities which is equivalent to the
system above. It is not hard to show that the following system satisfies our conditions:
x0  1, x1  1,
x2  1, x1 2  x2    x1 ,
x3  1, x1  x2  x3  x1  x2 ,
( x1  x3 )  x2 2  x4  ( x1  x3 )   x1 .
So, a procedure for computing all L-spectra of order 9 can be described. Now, using
Theorem 3 it is obvious that we can describe an effective method of constructing all Lgroups with skeleton G, where G is a cyclic group of order 9. This completes the proof.
For instance, let L be the standard Lukasiewicz structure, and let G  G, , 1, e be a
cyclic group of order 9 generated by a. The following algorithm produces all L-groups
G  G,  G , , 1, e , where  G is defined as a matrix [ cij ] 99 such that (a i  G a j )  cij ,
for all 0  i, j  8.
Algorithm. 1. Imputs: definitions of  and  in L, the Cayley table of the group G;.
s 3
2. Let s0 : 1; choose s1  [0,1); choose s2 such that s1 2  s2  1
;
4
3 Choose s3  [0,1) such that s1  s 2  s3  s1  s 2 ;


s 1

4 Choose s4  [0,1) such that max s1  s3 , s 2 2  s 4  min  s1  s3 , 1
;
2 

5 Let c0i : si , for all i, 0  i  4; put c0i : c0, 9i , for all i, 5  i  8; for all i, j,
1  i  8, 0  j  8, if i  j put cij : c 0, i  j , else put cij : c 0, j i ;
6 Outputs: the cyclic L-spectrum s0 , s1 , s2 , s3 , s4 of order 9, the matrix [ cij ] 99 , and
the L-group G,  G , , 1, e , where (a i  G a j )  cij , for all i, j, 0  i, j  8.
References
1. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517.
2. R. Bělohlávek, Fuzzy Relational Systems, Foundations and Principles, Kluwer
Academic/Plenum Publishers, New York, 2002.
3. R. Bělohlávek, Birkhooff variety theorem and fuzzy logic, Arch. Math, Logic 42
(2003) 781-790.
4. R. Bělohlávek, V. Vychodil, Algebras with fuzzy equalities, Fuzzy sets and systems
157 (2006) 161 – 201.
5. V. Vychodil, Direct limits and reduced products of algebras with fuzzy equalities, J.
Mult.-Valued Logic Soft Comput. 13 (2007), no 1-2, 1-27.
6. M. Demirci, J. Recasens, Fuzzy groups, fuzzy functions and fuzzy equivalence
relations, Fuzzy sets and systems 144 (2004) 441-458
7. M. Demirci, Fundamentals of M-vague algebra and M-vague arithmetic operations,
International Journal of Uncertanity, Fuzziness and Knowledge Based Systems 10 (2002)
25-75.
8. J. Recasens, Indistinguishability Operators, Modeling Fuzzy Equalities and Fuzzy
Equivalence Relations, Series: Studies in Fuzziness and Soft Computing , Vol. 260,
Springer Verlag, 2011.
9. M. Bradic, R. Madarász, Construction of finite L-groups, Fuzzy sets and systems,
submitted