Szpilrajn's theorem on fuzzy orderings

Fuzzy Sets and Systems 10 (1983) 101-108 North-Holland Publishing Company

SZPILRAJN’S Hiroshi Faculty Received Revised

THEOREM

101

ON FUZZY

ORDERINGS

HASHIMOTO of Economics.

Yamaguchi

Uniuersiry,

Yamaguchi

City,

753 Japan

September 1981 February 1982

Fuzzy relation matrices are considered and the well-known Szpilrajn theorem on orderings is generalized to fuzzy orderings. Zadeh has already shown a generalized theorem, but another generalization is given in matrix form. Some interesting operations on fuzzy matrices are introduced, and their properties are mentioned briefly. The theorem is represented in terms of the matrix operations. The operations have many properties and are useful in the various applications of fuzzy matrices or fuzzy relations. Keywords: Fuzzy relation, Symmetric

relation, kernel.

Fuzzy

matrix,

Fuzzy

ordering,

Szpilrajn’s

theorem.

Transitive

1. introduction We consider fuzzy relation matrices, and generalize Szpilrajn’s theorem on orderings [2,12]. The Szpilrajn theorem is a very well-known result. Zadeh [16] has already shown a generalized theorem, but we give another generalization in matrix form. In the paper some interesting matrix operations are defined, and their basic properties are examined briefly. The theorem is represented in terms of the matrix operations. The operations have interesting properties so that they are useful in the study of fuzzy matrices.

2. De&&ions Some operations and notations are defined. For x, y which take values in the unit interval [0, 11, we define x v y, x A y, x0 y as follows: xv y = max(x, y). x A y = min(x, y), xey

=

I

X

0

if x>y, if x < y.

For n x n fuzzy matrices R = [rii] and S = [sii], all of whose elements belong to 0165-0114/83/0000-0000/$03.00

@ 1983 North-Holland

102

H. Hashimoto

the unit interval,

we define the following

matrix operations:

RVS=[rijVSij],

RAS=[rijASij], ROS=[rijOSij],

RxS=

=[(r,lr\s,i)v(rizAszi)v..

R” = I = [Sii]

(Sii is the Kronecker

R k+‘=Rk~R

(k=0,1,2

R’=RvR2v...vR” R’ =[rii]

.v(r,,~s,~)],

delta),

,... ), (the transitive closure of R),

(the transpose of R),

AR = ROR’, VR = R

AR’

(the symmetric

kernel),

RGS , R
if and only if

rii G sii for all i, j,

if and only if

rii < sii for all i, j.

Above all, the operation 0 is essential to our study. We deal only with fuzzy matrices. Now we define some special types of matrices which play an important role in fuzzy matrix theory [lo, 11,161. A matrix, all of whose elements are zero, is called the zero matrix, and is denoted by 0. A matrix R such that R*< R is said to be transitive. A matrix R is said to be idempotent if R2 = R. Obviously any idempotent matrix is transitive. A matrix, all of whose diagonal elements are zero, is said to be h-reflexive. For any matrix R, AR is u-reflexive. An n x n matrix R such that R” = 0 is said to be nilpotent [8]. It is clear that if R’ = 0 for some positive integer 1, then R” = 0 for some integer m (1s m < n). (See the Appendix for a proof.) An n x n matrix R is said to be nilpotent of degree m if there exists some integer m (1 s m < n) such that RI”-’ # 0 and R” =O. Any nilpotent matrix is n-reflexive. If R is nilpotent, then VR = 0. The converse is, in general, false. Nilpotent matrices are of great importance [4,6,8]. Acyclic graphs are represented by nilpotent matrices. Acyclic graphs correspond to consistent systems. If VR ~1, then R is (perfectly) antisymmetric [5, 161. If VR = R, then R = R’, so that R is symmetric. Since AR = R 0 VR, if AR = 0 then R is symmetric. However it is not true that VR = R 8 AR. Clearly VR is symmetric and V(AR) = 0. If R is symmetric, then AR = 0. Thus A(VR) = 0. On the other hand, VR = 0 if and only if AR = R.

3. Results First we prove a theorem which is a generalization of the Szpilrajn’s theorem and is represented by the matrix operations defined above. Then we obtain a corollary from the theorem.

Szpilrajn’s theorem on fuzzy

orderings

103

Theorem. Let R = [rii] (0 6 rii =Z1) be an n x n fuzzy transitive matrix, and let pf q be integers such that 1 > rw 3 rqP. We define an n x n matrix T = [tii] whose elements tij are given by t.. = 1,

b ’

‘vi

{ rij

if

(i,

i)

=

h

q),

if

(i,

i)

#

(p,

91,

where b E [0, 11. Then an n x n matrix S defined by S = [s,J = T’ fulfills

(1) (2) (3) (4) (5)

the following conditions: s*ss, R cS, AR GAS, VR =VS, s, = b, and s,, = rqp.

Proof. (1) Since (T’)*-’ T’, we have S”C S. (See [7] for a proof.) (2) R s T s T’ = S. (3) Assume that rij 0 rii > sii 0.~. Let c = rij 0 rii. Then rii=c>O, Therefore

rji

<

C,

and

sii 2 c.

there exist integers I,, I,, &I,A t&l,A . . . A fh, 2 c,

, l,, E { 1,2,

. . , n} such that

where msn-1.

Without loss of generality assume that integers j, II, 12, . . , Z,,,, i are distinct from each other. Let &=j and lm+l = i. If 1, = p and Ia+, = q for some (Y, then since rij = c, we have rqP?=c, so that rw Z=c. Then rji 2 c, which contradicts the fact that rji < c. Furthermore if 1, # p or I,, , # q for every (Y, then rji 3 c, which contradicts the fact that rii CC. Hence we have rii 0 rii < sij 0 sji, i.e. AR c AS. (4) Since VR 0. Then there exist integers {l, 2,. , n} (m, 1
c

fik,A&,kzA’

“Afk,,jArjh,Af,l,h2A’

“Afh,i

to prove that VScVR. Let k,, k2, . . . , k,, h,, h2, . . . , h, E

>C.

Without loss of generality assume that integers i, k,, k2, . . . , k, are distinct from each other and that h,, h2,. . . , h,, i are distinct from each other. Let k. = h[+, = i and k,,,, = h,, = j. (a) Suppose that (k,, k,,,) = (p, q) for some (Y and (hp, hptl) = (p, q) for some /3. Then since for some (Y,0

104

H. Hashimoto

we have rq,, > c, so that rrxl 3 c. Then rii A rii 2 c. (b) Suppose that (k,, k,+r) = (p, q) for some CYand (h,, h,,,) # (p, q) for every p. Then WC have r9,,a C, SO that rp, *c. Then ri, A rii 2 C. (c) Suppose that (k,, k,+,) #(p, q) for every (Y and (hp, hp+l) = (p, q) for some /3. Then we have rqP 3 C, SO that rN > C. Then rij A rii 3 C. (d) Suppose that (k,, k,,,) #(P, q) for every (Y and (hp, hp+J #(P, q) for every 0. Then rij A rji 2 c. Thus VSsVR. Hence VR =VS. (5) Since Tc S, we have s, 2 b. If s, > b, then there exist integers kl, kZ, . . . , k,, E {1,2, . . . , n} (m 2 1) such that

Without loss of generality assume that integers p, kl, k,, . . , k,,,, q are distinct from each other. Therefore rw > b, which contradicts the fact that rw --cb. Hence we have s, = b. Now since rw A rqp = rqp, by (4) we have s, AS,, = rqp. Hence S 4P= rqP. Cl Example.

Let

Then

AR=REIR’=

[

0 0.4 0 0 00

1

0.3 0.2 , 0

Setting p = 2, q = 3, and b = 0.4 we construct t-,= 1, Thus

a matrix

b=0.4>rz3=0.2

if (i,j)=(2,3),

rij

if (i, i) # (2-3).

T = [tij] as follows:

Szpilrajn’s

T3=

[“,’ 0.1

theorem

“,’ 0 1

0.J 0.1

x

[“,’ 0.1

7 01

04 oJ

X [y0.1

on fuzzy

orderings

105

“: 0.1

OJ [T %’ 0 4 = 0 1 0.1

r 01

OJ 04

91’1, 0.1

Moreover s2= [; 01

ASf

y

;:I$

vsf

= [“,’0.1

9:

r 0.1

01 :I,

i].

Therefore

S’
R
ARSAS,

Now we notice the following

and

VR=VS.

relationships:

As shown in this example, if R is transitive, then AR is irreflexive and transitive, so nilpotent. Furthermore VR is symmetric and transitive, so idempotent. These facts are shown in the Appendix. Using the theorem,

Corollary. (1) (2) (3) (4) (5)

we have the following

corollary.

If R is transitive, then there exists a matrix S such that

s2cs,

RsSS, AR< AS, VR =VS, O
106

H. Hashimoro

Proof. (l)-(4) Obvious. (5) If rw = rqp = 0 (p # q), we can construct a matrix S such that s, = 6 > 0. and s4p =0 using the theorem. Repeating this argument, we have O
cl

As is well known, the Szpilrajn theorem states that any partial ordering on a set can be extended to a total ordering [l, 121. A partial ordering is reflexive, asymmetric, and transitive. A total ordering is reflexive, asymmetric, transitive, and connected [3, 141. Arrow [2] states the Szpilrajn theorem as follows: There exists a weak ordering compatible with any quasi-ordering. A weak ordering is reflexive, transitive, and connected. A quasi-ordering is reflexive and transitive. Given two boolean relations R, S if (a) xRy implies xSy for all x, y and (b) xRy and not yRx do not imply ySx for all x, y, then S is said to be compatible with R [2]. Considering R and S to be boolean matrices and using our notations, if R O, then pL(a(x), a(y)) = pP(x, y), x, y E X. We generalize Arrow’s representation, and our results are considered to be useful in the discussion of fuzzy preferences [2, lo]. The results give a detailed structure of an extention of partial orderings. An advantage of the matrix representation is that the representation is compact and clear. We can construct a new ordering using matrix operations. Furthermore the invariance of the symmetric kernel in the extention is shown explicitly.

4. Concluding

remarks

We represent a generalized Szpilrajn theorem using matrix operations A and V. These operations have many properties and are useful for the various applications. For example, they are very useful in the discussion of preference relations, retrieval models, and so on [lo, 13, 151. We can represent basic propositions of the theory of preference relations in terms of matrix operations. Furthermore we can deal with the relations in the matrix form. If a matrix R is reflexive and transitive, then VR is a matrix which represents an equivalence relation or a similarity relation [lo, 161. Using VR, we can obtain certain classes which are useful for the construction of a thesaurus in information retrieval [9].

Appendix Proposition. Let R = [rii] be an n x n fuzzy matrix and let Rk = [$!“I. (1) If R’ = 0 for some positive integer I, then R” = 0. (2) If R is transitive, then AR is transitive.

Szpilrajn’s

(3) (4) (5) Proof.

If If If

theorem

on fuzzy

orderings

107

R is transitive, then OR is transitive. R is transitive and symmetric, then R is idempotent. R is transitive, then VR is idempotent.

(1) Suppose that R”# 0. Then rik,

A rk,k2

A * ’ ’ A rkH ,i #O

for some integers i, k,, kZ, . . , k,-l, j E {1,2, k, = k, for some o! < 0. We have ’

. . , n}. Let k, = i and k,, = j. Hence

rlpi”a ’ # 0 so that R”“-“’

f 0.

This is a contradiction. Therefore, R” = 0. (2) Let S =[Sii]= AR. Then Si; = rii 0 rj,. Suppose that rik A rkl = c

and

r,, L c.

We show that. if ric2 r+ then there exists a contradiction. (a) If rik = c, then rki Cc. However rki 2

rkj

A rii 2 C.

This is a contradiction. (b) If rki = c, then rik CC. However rik 2 rii A rik 2 c. This is a contradiction.

Thus we have rii
sii = rii 3 c. (3) Since R AR’S R and R AR’S (R/TR’)~sR~cR Therefore,

(R

A

R’)2 S R

and A

R’, we have (RAR’)*s(R’)~sR’.

R’, i.e.

(VR)‘sVR. (4) Let rii = c 2 0. Then rii = c,

SO

that

rii > rii A rii = c. On the other hand r!;’ 2 rii A rii = c. We have c =Srif S rii = c. Then rf’ = c and hence R2 = R. (5) Since VR is symmetric, by (3) and (4) we have (VR)’ = VR.

0

sik

A ski = c >O. We have

H. Hashimoto

108

References [I] [2] [3] [4] [S] [6] [7] [8] [9] [IO] [II] [12] [13] [14] [15] [16]

M. Aigner, Combinatorial Theory (Springer, New York, 1979). K.J. Arrow, Social Choice and Individual Values (2nd edn.) (Wiley, New York, 1963). B. Car& Graphs and Networks (Clarendon Press, Oxford, 1979). N. Deo, Graph Theory with Applications to Engineering and Computer Science (Prentice-Hall, Englewood Cliffs, NJ, 1974). D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980). F. Harary, On the consistency of precedence matrices, J. ACM 7 (1960) 255-259. A. Kaufmann. Introduction to the Theory of Fuzzy Subsets Vol. 1 (Academic Press, New York, 1975). R.B. Marimont, Applications of graphs and boolean matrices to computer programming, SIAM Rev. 2 (1960) 259-268. C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis (Birkhauser Verlag, Basel, 1975). S.V. Ovchinnikov. Structure of fuzzy binary relations, Fuzzy Sets and Systems 6 (1981) 169-195. D. Rosenblatt, On the graphs of finite idempotent Boolean relation matrices, J. Res. Nat. Bur. Standards 67B (1963) 249-256. E. Szpilrajn, Sur I’extension de I’order partiel, Fundamenta Mathematicae 16 (1930) 386-389. V. Tahani, A fuzzy model of document retrieval systems, Inform, Process. Managem. 12 (1976) 177-187. A. Tarski, Introduction to Logic (Oxford University Press, New York, 1965). W.M. Turski, On a model of information retrieval system based on thesaurus, Inform. Stor. Retr. 7 (1971) 89-94. L.A. Zadeh, Similarity relations and fuzzy orderings, Inform. Sci. 3 (1971) 177-200.