Fuzzy Sets and S@tems 29 (i989) ~\'9--234 North-Holland

229

TP~AI~SITIIVI[TY OF F U Z Z Y M A T ~ C E S U N D E R G E N E R A L I Z E D C O . N I ~ ~ D N E S S

Hiroshi HASHIMOTO Facully of Economics, Yamaguchi University, YamaSuchi 753, Japan

Received October 1986 Revised April 1987

Absgr¢,ct: Generalizing the ordinary connectedness of binary relations we show some properties of transitive fuzzy matrices under generalized connccted='ss. That is, introducing a special type of subtra~,m operation we obtain several conditions for a given furry ~atrix to become transitive ~nder ¢on~,~ectednes~. Obtained results are generalizations of propositions on Boolean tr~atrices representing tournaments which are asymmetric and weakl? connected relations, or asymmetric and complete digraphs. They are considered to be useful for discussions of fuzzy preferences, fuzzy digraphs, and so on. The generalized propositions hold for Boolean matrices as well and as a special case we can have properties of ordinaw tournaments.

Keywords: Fuzzy matrix; f~!,~y rela~'~ion; fuzzy digraph.

1. ]ntreduction

We consider matrices representing transitive generalized connected fuzzy relations and obtain some properties of transitive fuzzy matrices. Transitive f u r y relations are esseatial in many applications of fuzzy ,~et theory [2, 8]. The results obtained generalize some properties of Boolean matrices representing tourna- ments (asymmetric and weakly connected relations [3]). As a special case we have some properties of ordinary tournaments. Tournaments, especially transitive tournaments, have many interesting properties and are important in practical applications [1, 7].

In the paper we introduce a special type of subtraction operation and using this ope~'ation we examine prol~erties of transitive connected fuzzy matrices. These properties of fuzzy matrices show the relationship between transitivity, anti- symmetry (i.e., the corresponding fuzzy digraphs are 2-cycle free), nopexistence of cycles of length 3, and nilpotency. Thus the results are closely related to fuzzy preferences, fuzzy digraphs, and so on.

2. Results

In what follows we adopt the usual operations on fuzzy matrices except subtraction. About subt~'action we introduce a special type of operation and show a basic proposition on f~,.zy matrices. Then we derive equivalent conditions for

0165-0114/89/$3.50 (~) 1989, Elsevier Science Publishers B.V. (North-Holland)

2~0 h: l-lash~moto

fuzzy matrices from the proposition. Finally we show several conditions for a given f u r y matrix to be transitive and obtain some properties of ordinary transitive tournaments as a special case.

Defufifion 1. For x, y e [0, 1] put x v y = max(x , y) , x ^ y = rain(x, y) . The subtraction • is an arbitrary n.x¢~ H~:ary operation (groupoid) on [0, 1] satisfying x * 0 = x and x* y <~ x for a]! x, y ~ [0, 1]. Let n be a fixed positive integer and F the set of n x n fuzzy matrice~ (matrices over [0, 1]). The oper~i~-~ v~ ~ , are defined componentwise (e.g. for R = [rij], S - [$ij] E F, R * 5 --" [rij* $ijJ). Further we put

R x S = (r~ ^ s~ ) , R ~ = R ~- ~ × R (k >~ 2).

The order <~ on F is defined componentwise, i.e.

R <~S if r~/ <<-s~ fi)r all i, j.

We say that R is transitive if R 2 ~< R. The transpose of R is den,)ted R T and 0 and I stand for t[~e zero and identity n x n matrices. We say that R is antisymmetric if R ^ R T <~ I [8].

Peoposltion 1. I f to R ~ F there i.~ S (R <<. S e F) such that

( R * S T ) 2 < ~ R v R T v [ , R A S T ~ [ , ( R * S T ) ~ ^ I = O ,

th~. R ~ o n ~ y m m e t r i c and tra~siti,e.

Proof. The conditions R ^ S T ~< L R ~< S imply that R A R r <~ R A S T ~< 1 (i.e. R i: antisymmetric). Put U = [u~j] ffi R . S "r. We have uij = r 0 for ~ ~ ] . ( Indeed, u o ffi r:j ~o sj~ ~< ~7 and so u~j = r~j if rtj ffi 0. If r~ ~ 0 then s~ = 0 and u~j = r~j • 0 ffi r4j.) We prove R 2 ~< R. Let i ~.i. In view of ra~ A rij <<. r~ i and r~ i ^ ,-j:~ <. rij it suffices to check r~k ^ rkj <~ r~/for all i # k ~ j . Here r~k ffi u~k and r#i = ukj ~nd so we check u~k ^ ukj ~ r o. (i) Suppose r~j ~ 0. Then r# ffi 0 and from U 2 ~< R v R T v I we get the required u~k ^ uk~ ~< r~j v r# ffi r~ r (ii) Thus let rs~ = 0. If r~ = 0 then from U 2 ~< R v R T u I we get u~k ^ u~ ffi 0 and we are done. Thus let rj~ ~ 0. Using U 3 ̂ I - / ) we get

0 = u,k ^ u~ ^ u~ = (u~ ^ u ~ ) ^ r~

and so again u~ ^ u~ i ffi 0. This settles the case i ~ j . Since R is antisymmetric we have r~ ^ r~ ffi 0 ~< ~ for all k ~ i and r~/ , r~ ffi r,~, which concludes the proof. []

Exmnp|e L Let x * y - rain(x, Ix - y + O.lJ),

[004 dO 1 Then

R,ST [ ?]

Transitivity of fuzzy matrices

Lemma 1. I f R ^ S 7 <~ 1, then (R * S T) v (R ^ I) -- R.

Proof. Obvious. 1"3

I[~OpeS|fiOU 2. I f R<~S, (R*ST)2<.R v R T v l, I~ AST<~I, (R*ST)a ^ I = O , then R2 <~ (R * S T) V (R /, I).

Proof. By Proposition 1 and Lemma 1. []

Defu~tion 2, x e y is an operation on [0, 1] satisfying (1) x ~ O = x , (2) x <~ y implies x e y = O, x e z ~< y e z, z e y <~ z G x for all x, y, z E [0, 1].

No~e that since 0~<y, we have x e y ~ x by (1) and (2).

Ezmnple 2.

(1) x e y f f i { 0 - y ifx > y , i fx ~<y.

Ix i fx > y , (2) x e y = ~ O i f x < y .

Wz de, fine R e s componcntwise for fuzzy matrices R and S.

Proposition 3 [4, 5, 6]. I f R 2 <~ R, then (R e RT) 2 <~ R ~ R T.

C o r d m T 1.

Proof. Since .~t~ 3 A I ffi O.

I f R2 <~ R, then (R e RT)3 A I ffi O.

R e R T is nilpotent, that is, (R e / { r). ffi 0 [4], []

we have (R e

Lemma 2, If R2<.(R O RT) v (R h i ) , then R A R'r ~;.

Proof. Assume that re ^ rja ~ 0. Let R z ffi [r~)]. Since r~ 2) 4:0 and (R e R T) ^ I = 0, we have r~s ~ 0, so r~ ) ~ 0, r~ 2) ~ 0. Then since R e R T is antisymmetric, we have i -- ] from R 2 ~< (R e R T) v (R ^ I). Hence R A R T ~< I. []

Lemma 3. For any fuzzy matrix R, (1) R A R T <~ L (2) (R e I) 2 ^ ~ = 0. (3) R e l f f i R e R T.

the following conditions are e~uivMent:

r [oeL Obvious. []

Proposition 4. I f (R e RT) e ~< R v R x v I, then the following conditions are

232 H. Hashimoto

equivalent: (1) R ^ R T < ~ I , ( R e R T ) 3 ^ I = O . (2) ((R O 02 V (U e RT) 3) ^ I --" 0.

((R e I) v (R e 0 3) ^ l = o. (4) R 2~<(ROR T) V(R ^!) .

Proof. (1) implies (2). Obvious by Lemma 3. (2) implies (3). Since (R O I ) 2 ^ 1 = 0, we have R e I = R O R T by Lomma 3,

so (R O 1) 3 ̂ I = 0. (3) implies (4). Then R ;, R T ~< I by Lemma 3 and since R e R T ~< R e I, we

have (R e R T ) 3 A I -- 0, SO by Proposition 2,

R 2 ~< (R e R T) V (R ^ I).

(4) replies (1). Clearly R 2 ~< (R e R r) v (R ^ 1) <~ R. By Corollary 1 we have (R O R ~ 3 ̂ I = 0 and by Lcmiaa 2 R A R T ~ I

Now we have two corollaries on an n x n fuzzy matrix R from the above p~Jl',osition as follows.

CoroUalry 2° I f (R 0 RT)2<--.R v R T v I, then: (1) ((R e 1) 2 v (R 0 I) 3) ^ 1 =0 implies R2<~R, (2) (R e I) 6 ̂ I = 0 implies R 2 <~ R, (3) (R e 1) n = 0 implies R 2 <~ R, (4) (R 2 v R 3) ^ I = 0 implies R 2 <~ R, (5) R 6 A 1 = 0 implies R 2 <<. R, (6) R" = 0 implies R 2 <~ R.

Proof. (1) By Proposition 4, R2~ (R e R T) v (R ^ 1)~ < R. (2) Since (R e !)6A ! =0, we have (R e 1) 2 ̂ 1 = 0 and (R e l) s ^ I =0. Then

R2<~R by (1). (3) Since (R e 1)" - 0 , then (R e I) e" - 0 , so (R e 1) 6 ^ I =0 . By (2) we have

RZ~R. (4)-(6) Obvious by (1)-(3). []

Corol~d 3o I f R v R T v l - J , the universal matrix, then: (1) ((R e l )2v (R e 1) 3) ^ 1 = 0 implies R 2 ~ R , (2) (R e I) 6 ,x 1 = 0 implies R 2 <~ R, (3) (R ~ I) n = 0 implies R 2 <~ R, (4) (R 2 v R 3) ^ I = 0 implies R 2 <~ R, (5) R 6 ̂ 1 = 0 implies R 2 <<. R, (6) R n = 0 implies R 2 <<- R.

Clearly if R v R T v l = Y and R 2 ^ I = 0 (or R ^ R r = 0 ) , then R is Boolean. Therefore the R's in Corollary 3, (4)-(6) are Boolean. Thus the matrix R which fulfills R v R T v 1 --,r and R 2 ̂ 1 = 0 represents an ordinary tournament. These properties of tournaments in Corollary 3, (4)-(6) are well known [1, 7].

Trans~dv~y of ~zzy mo~es 233

In the theory of ordinary relations if R v R T V 1 = J then a relation represented by the Boolean matrix R is called weakly connected and if R v R T = J then connected [3]. Therefore a tournament is a digraph corresponding to a weakly connected and asymmetric (R ^ R T = 0 ) relation. Properties of a transitive tournament were studied in many works [1, 3, 7]. We generalized the condition of weak connectedness R v R T v 1 ffi ? to R e R T ~ R v R T v I, and under the generalized condition we obtained some properties of transitive fuzzy matrices in Corollary 2 and Corollary 3 as a special case. The following proposition is related to connectedness.

Proposition $. I f R v R T - - Y ~nd (R e I ) 3 ^ 1 = O, then R is idempotent, that is, R 2 f R .

Proof. Since I <~ R, it is sufficient to show that R 2 ~ < R. Assume that rt~ ^ rkj ffi C>0 .

Case 1. i ffi k. Then ro = rk~ >~ c. Case 2. k = ]. Then r~j ffi r~k ~> c. Case3. i = j. Then ~jfr~jffil~>c. Case 4. i ~ k, k ~] , i ~] . Since rjk ~ c, rk~ ~> c, and (R e 1) 3 ~ 1 = 0, we have

rj~ ffi O. Therefore r~ = 1 ~> co []

Example 3. Let

[ 1 1 1 1

R = 0 6 1 1 .

0 0 1

L!~en R ,7 R T ffi Y, (R ~ I ) 3 ^ 1 ---- 0, and R 2 "- R.

3. Conclusions

We showed some properties of transitive fuzzy matrices u~de~ ~he generalized connectedness. That is, we obtained several conditions for a given fuzzy matrix t~ become transitive. The results are useful for discussions of fuzzy preferences and fuzzy digraphs. Our results hold for Boolean matrices as well. Connected transitive fuzzy matrices have many interesting properties. It is t~n imp~rtsnt problem to examine prop,:rties of connected fuzzy matrices in respc~:t of ~,~rve relationship between nilpotency and transitivity. Further we could generalize out results for fuzzy matrices over [0, 1] to g~meralized fuzzy matrices, e.g., to matrices over a bounded chain.

Acknow|edgments

The author is indebted to the referees for their constructive remarks in revising the paper, especially Proposition 1 and its proof.

234

Ref rences

H. Hashimoto

[1] M qehzad, G. Chartrand and L. Lesniak-Foster, Graphs and Digraphs (Wadsworth, Belmont, CA, 1779).

[2] D. Dubois and Ho Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980).

[3] P.C. Fishburn, The Theory of Social Choice (Princeton University Press, Princeton, NJ, 1973). [4] H. Hashlmoto, Transitivity of generalized fuzzy maffices, Fuzzy Sets and Systems 17 (1985) 83-90. [5] S.A. Oriovsky~ Decision-m_aHng with a fuzzy preference relation, Fuzzy Sets and Syslems 1 (1978)

155-167. [6] S.V. Ovchinnikov, Structure of fuzzy binary relations, Fuzzy Sets and Systems 6 (1981) 169-195. [7] F.S. Roberts, Discrete Mathematical Models, With Applic~m'o~zs to Social, Biological, and

Environmental Problems (prentice.~all, Englewood Cfiffs, N J, 1976). [8] L.A. Zadeh, Similarity relations attd fuzzy orderings, Inform. Sci. 3 (1971) 177-200