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<ul><li><p>Vague groups and generalized</p><p>concept are studied. The concept of external direct product of vague groups is estab-</p><p>* Tel.: +90 242 3102360; fax: +90 242 2278911/3102360.</p><p>E-mail address: sevdasezer@akdeniz.edu.tr</p><p>Information Sciences 174 (2005) 123142</p><p>www.elsevier.com/locate/inslished. Later the denition of generalized vague subgroup, which is a generalization</p><p>of the vague subgroup dened by Demirci, is introduced, and the validity of some clas-</p><p>sical results in this setting is investigated on the basis of the particular integral commu-</p><p>tative, complete quasi-monoidal lattice ([0,1], 6 ,^). 2004 Elsevier Inc. All rights reserved.</p><p>Keywords: Fuzzy equality; Fuzzy function; Vague group; External direct product of vague groups;</p><p>Vague homomorphism; Vague isomorphism; Vague subgroup; Generalized vague subgroupvague subgroups on the basis of ([0,1], 6 ,^)Sevda Sezer *</p><p>Department of Mathematics, Faculty of Science and Arts, Akdeniz University, 07058 Antalya, Turkey</p><p>Received 2 July 2004; received in revised form 21 July 2004; accepted 24 July 2004</p><p>Abstract</p><p>In this paper, various elementary properties of vague groups and some properties of</p><p>vague binary operations related with their associativity aspects are obtained. Further-</p><p>more, the concept of vague isomorphism is dened and some basic properties of this0020-0255/$ - see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.ins.2004.07.016</p></li><li><p>nitely many real numbers, and X, Y, G will always stand for crisp and non-empty sets in this paper.</p><p>Denition 1 [1]. A mapping EX : X X! [0,1] is called a fuzzy equality on Xif the following conditions are satised:</p><p>(E.1) EX x; y 1() x y, "(x,y) 2 X,(E.2) EX(x,y) = EX(y,x), "x,y 2 X,(E.3) EX(x,y) ^ EX(y,z) 6 EX(x,z), "x,y,z 2 X.</p><p>For x,y 2 X, the real number EX(x,y) shows the degree of the equality of xand y. One can always dene a fuzzy equality on X with respect to (abbreviated</p><p>to w.r.t.) the classical equality of the elements of X. Indeed, the mapping</p><p>EcX : X X ! 0; 1, dened byEcX x; y :</p><p>1; x y;0; x 6 y</p><p>is obviously a fuzzy equality on X.1. Introduction</p><p>After the introduction of fuzzy sets by Zadeh [14], fuzzy settings of various</p><p>algebraic concepts were studied by several authors, using the approach of</p><p>Rosenfeld [11]. In Rosenfelds paper, only the subsets are fuzzy and the groupoperation remains crisp. To get a more general extension, Demirci [2] denedthe concept of vague group based on fuzzy equalities and fuzzy functions</p><p>given in [1,3]. In the same paper, the concepts of vague subgroup and vague</p><p>homomorphism were dened and the validity of some classical results in this</p><p>setting were investigated. Later, some characterizations and examples concern-</p><p>ing vague groups were given by Demirci and Coker [4].</p><p>The general theory of vague algebraic notions as well as vague groups and</p><p>vague subgroups has been established in [68] based on a xed integral commu-</p><p>tative, complete quasi-monoidal lattice (for short, cqm-lattice) (L, 6 ,w). Thiswork introduces some elementary properties of vague groups and generalized</p><p>vague subgroups, and establishes some new results on the basis of the parti-</p><p>cular integral, commutative cqm-lattice ([0,1], 6 ,^).</p><p>2. Preliminaries</p><p>The symbol ^ will always stand for the minimum operation between</p><p>124 S. Sezer / Information Sciences 174 (2005) 123142Denition 2 [5]. Let EX and EY be two fuzzy equalities on X and Y,</p><p>respectively. Then a fuzzy relation ~ from X to Y (i.e., a fuzzy subset ~ of</p></li><li><p>Denition 3 [2,5](i) A strong fuzzy function ~ : X X,X w.r.t. a fuzzy equality EXX onX X and a fuzzy equality EX on X is called a vague binary operationon X w.r.t. EX X and EX. (For all (x1,x2) 2 X X, x3 2 X,l~x1; x2; x3 will be denoted by l~x1; x2; x3 for the sake of simplicity.)</p><p>(ii) A vague binary operation ~ on X w.r.t. EXX and EX is said to be tran-sitive of the rst order if l~a; b; c ^ EX c; d6 l~a; b; d for alla,b,c,d 2 X.</p><p>(iii) A vague binary operation ~ on X w.r.t. EXX and EX is said to be tran-sitive of the second order if l~a; b; c ^ EX b; d6 l~a; d; c for alla,b,c,d 2 X.</p><p>(iv) A vague binary operation ~ on X w.r.t. EXX and EX is said to be tran-sitive of the third order if l~a; b; c ^ EX a; d6 l~d; b; c for alla,b,c,d 2 X.</p><p>3. Vague groups</p><p>3.1. Denition of vague groups</p><p>The concept of vague group, which is the base of this work, is given by the</p><p>following denition.</p><p>Denition 4 [2]. Let ~ be a vague binary operation on G w.r.t. a fuzzy equalityEGG on G G and a fuzzy equality EG on G. Then</p><p>(i) G together with ~, denoted by hG; ~;EGG;EGi or simply hG; ~i, is called aX Y) is called a strong fuzzy function from X to Y w.r.t. the fuzzy equalitiesEX and EY, denoted by ~ : X,Y , if the characteristic functionl~ : X Y ! 0; 1 of ~ satises the following two conditions:</p><p>(F.1) For each x 2 X, there exists y 2 Y such that l~x; y 1,(F.2) For each x1,x2 2 X, y1,y2 2 Y,</p><p>l~x1; y1 ^ l~x2; y2 ^ EX x1; x26EY y1; y2:</p><p>The concepts of vague binary operation on X and transitivity of a vague bin-</p><p>ary operation are dened as follows.</p><p>S. Sezer / Information Sciences 174 (2005) 123142 125vague semigroup if the characteristic function l~ : G G G! 0; 1 of~ fullls the condition: For all a,b,c,d,m,q,w 2 G,</p></li><li><p>l~b; c; d ^ l~a; d;m ^ l~a; b; q ^ l~q; c;w6EGm;w:(ii) A vague semigroup hG; ~i is called a vague monoid if there exists a two-</p><p>sided identity element e 2 G, that is an element e satisfyingl~e; a; a ^ l~a; e; a 1 for each a 2 G.</p><p>(iii) A vague monoid hG; ~i is called a vague group if for each a 2 G, thereexists a two-sided inverse element a1 2 G, that is an element a1 satis-fying l~a1; a; e ^ l~a; a1; e 1.</p><p>(iv) A vague semigroup hG; ~i is said to be commutative (Abelian) ifl~a; b;m ^ l~b; a;w6EGm;w for each a,b,m,w 2 G.</p><p>In particular, if ~ is a vague binary operation on G w.r.t. EcGG on G G andEcG on G such that l~G G G f0; 1g, then a vague group hG; ~icorresponds in a one-to-one way with a group in the classical sense. In this</p><p>case, a vague group is simply called a crisp group. For a given classical group</p><p>hG,i, it is known that an innite number of nontrivial vague groups can bedened on G [2,4,5].</p><p>In the rest of this paper, the notation hG; ~i always stands for the vaguegroup hG; ~i w.r.t. a fuzzy equality EGG on G G and a fuzzy equality EGon G.</p><p>Proposition 5 [2]. For a given vague group hG; ~i, there exists a unique binaryoperation in the classical sense, denoted by , on G such that hG,i is a group inthe classical sense.</p><p>The binary operation in Proposition 5 is explicitly given by theequivalence</p><p>a b : c() l~a; b; c 1; 8a; b; c 2 G: 1The binary operation , dened by the equivalence (1), is called the ordinarydescription of ~, and is denoted by ord~ in [5,7,8].</p><p>If ~ is a vague binary operation on G w.r.t. a fuzzy equality EGG on G Gand a fuzzy equality EG on G, in the rest of this paper the ordinary description</p><p>of ~ will be denoted by . In this case, from [5,7] we have the followingproperty:</p><p>l~a; b; a b 1 and l~a; b; c6EGa b; c; 8a; b; c 2 G: 2If hG; ~i is a vague semigroup, then hG,i is a semigroup [5].</p><p>Conditions under which the converse holds are given below.</p><p>Proposition 6. Let ~ be a vague binary operation on G w.r.t. EGG and EG. If</p><p>126 S. Sezer / Information Sciences 174 (2005) 123142hG,i is a semigroup and ~ is transitive of the second and third orders, then hG; ~iis a vague semigroup.</p></li><li><p>Proof. If hG,i is a semigroup, then a (b c) = (a b) c for each a,b,c 2 G.Since ~ is transitive of the second and third orders then for alla,b,c,d,m,q,w 2 G we have</p><p>l~b; c; d ^ l~a; d;m ^ l~a; b; q ^ l~q; c;w6EGb c; d ^ l~a; d;m ^ EGa b; q ^ l~q; c;w6l~a; b c;m ^ l~a b; c;w6EGa b c;m ^ EGa b c;w6EGm;w:</p><p>Hence, hG; ~i is a vague semigroup. hIf ~ is a vague binary operation on G, hG,i is a semigroup and ~ is not tran-</p><p>sitive of the second and third orders then hG; ~i may not be a vague semigroup.The following example illustrates this case.</p><p>Example 7. Take the set G : {0,1,2}, and consider the fuzzy equalityE = [E (i, j)] on G by means of the matrix</p><p>and the fuzzy equality EGG : EcGG on G G. Now let us dene the fuzzyfunction ~ : G G,G by the matrix l~ l~x; i; j333, where x, i, j = 0,1,2(i: rows, j: columns) and</p><p>1 .5 1 .4</p><p>2 .4 .4 1</p><p>S. Sezer / Information Sciences 174 (2005) 123142 1270 1 2</p><p>l~0; i; j0 1 .5 .4</p><p>1 .5 1 .4</p><p>2 .4 .4 1</p><p>l~1; i; j0 .3 1 .21 .1 .1 1</p><p>2 1 .1 .2</p><p>l~2; i; j0 .3 .3 1</p><p>1 1 .2 .1G G 33</p><p>EG(i, j) 0 1 2</p><p>0 1 .5 .42 .2 1 .2</p></li><li><p>a,b,y 2 G.(ii) EG(a,b) = EG(y a,y b) for all a,b,y 2 G.</p><p>(c) If the vague binary operation ~ is transitive of the second and third orders,then EG(a,b)^EG(c,d) 6 EG(a c,b d) for all a,b, c,d 2 G.</p><p>Proof(a) (i) By the third-order transitivity of ~, for all a,b,x 2 G we haveSince</p><p>l~1; 0; 1 ^ EG0; 1 1 ^ :5 :5il~1; 1; 1 :1and</p><p>l~0; 0; 0 ^ EG0; 1 1 ^ :5 :5il~1; 0; 0 :3;~ is not transitive of the second and third orders on G. Furthermore, hG; ~i alsois not a vague semigroup because</p><p>l~0; 1; 0 ^ l~0; 0; 0 ^ l~0; 0; 1 ^ l~1; 1; 2 :5iEG0; 2 :4:</p><p>3.2. Some results on transitivity</p><p>The following theorem states that some results of classical algebra are also</p><p>valid for the vague algebra.</p><p>Theorem 8. Let hG; ~i be a vague group and e an identity element of hG; ~i.</p><p>(a) If the vague binary operation ~ is transitive of the third order, then(i) EG(x a,b) = EG(x,b a1) and EG(x a, e) = EG(x,a1) for all</p><p>a,b,x 2 G.(ii) EG(a,b) = EG(a x,b x) for all a,b,x 2 G.</p><p>(b) If the vague binary operation ~ is transitive of the second order, then(i) EG(a y,b) = EG(y,a1 b) and EG(a y,e) = EG(y,a1) for all</p><p>0 0 1 21 1 2 1</p><p>2 2 0 1 0 1 2Under these selections, ~ is a vague binary operation on G and we get hG,i as</p><p>128 S. Sezer / Information Sciences 174 (2005) 123142EGx a; b EGx a; b ^ l~x a; a1; x6l~b; a1; x6EGb a1; x:</p></li><li><p>On the other hand,</p><p>EGb a1; x EGb a1; x ^ l~b a1; a; b6 l~x; a; b6EGx a; b;</p><p>i.e., EG(x a,b) = EG(x,b a1). In particular, if b = e then, we observethat EG(x a,e) = EG(x,a1).(ii) By the third-order transitivity of ~, we haveEGa; b EGa; b ^ l~b; x; b x6 l~a; x; b x6EGa x; b xand</p><p>EGa x; b x EGa x; b x ^ l~b x; x1; b6l~a x; x1; b6EGa; b;</p><p>for all a,b,x 2 G. So EG(a,b) = EG(a x,b x).(b) (i) By the second-order transitivity of ~, for all a,b,y 2 G we have</p><p>EGa y; b EGa y; b ^ l~a1; a y; y6 l~a1; b; y6EGy; a1 b:</p><p>On the other hand,</p><p>EGy; a1 b EGy; a1 b ^ l~a; a1 b; b6 l~a; y; b6EGa y; b;</p><p>i.e., EG(y,a1 b) = EG(a y,b).</p><p>In particular, if b = e, then we have EG(a y,e) = EG(y,a1).(ii) By the second-order transitivity of ~, we have the inequalities</p><p>EGa; b EGa; b ^ l~y; b; y b6 l~y; a; y b6EGy a; y b;</p><p>EGy a; y b EGy a; y b ^ l~y1; y b; b6 l~y1; y a; b6EGa; b</p><p>for all a,b,y 2 G, thus EG(a,b) = EG(y a,y b).(c) By the second and third order transitivity of ~, we have</p><p>EGa; b ^ EGc; d l~a; c; a c ^ EGa; b ^ l~b; d; b d ^ EGc; d6 l~b; c; a c ^ l~b; c; b d6EGa c; b d; 8a; b; c; d 2 G: </p><p>Some inequalities about associative properties of vague semigroups are ob-</p><p>tained in the following two propositions.</p><p>Proposition 9. Let ~ be a vague binary operation on G w.r.t. fuzzy equalitiesEGG on G G and EG on G such that ~ is transitive of the third order. For</p><p>S. Sezer / Information Sciences 174 (2005) 123142 129 ord~, let hG,i be a semigroup. For n P 2, a1,a2, . . ., an, u1,u2, . . ., un 2 Gand u1 = a1, the following inequality is satisfied:</p></li><li><p>required inequality is true for n = 2. We assume that the required inequality</p><p>n^2l u ; a ; u 6E a a a ; u :</p><p>n^2</p><p>The aim of this subsection is to establish the concept of external direct prod-uct of vague groups. First, let us introduce the concept of the product fuzzy</p><p>equalities.</p><p>Let E and E E be fuzzy equalities on G and G G , respectively, fori1</p><p>l~ui; ai1; ui1 ^ l~un1; an; un</p><p>6EGa1 a2 an1; un1 ^ l~un1; an; un6l~a1 a2 an1; an; un6EGa1 a2 an; un: </p><p>Proposition 10. Let ~ be a vague binary operation on G w.r.t. fuzzy equalitiesEGG on G G and EG on G such that ~ is transitive of the second order. Let ord~ and hG, i be a semigroup. For n P 2, a1,a2, . . ., an, u1,u2, . . ., un 2 Gand u1 = a1, the following inequality is satisfied:</p><p>n^1</p><p>i1l~ai1; ui; ui16EGan an1 a1; un: 3</p><p>Proof. In a similar fashion to Proposition 9, the assertion can be easily veried</p><p>by applying induction on n, and the details are omitted. h</p><p>3.3. External direct product of vague groupsi1~ i i1 i1 G 1 2 n1 n1</p><p>Using this inequality and the third order transitivity of ~, we obtain theinequalities</p><p>n^1</p><p>i1l~ui; ai1; ui1 !is true for n 1, i.e.n^1</p><p>i1l~ui; ai1; ui16EGa1 a2 an; un:</p><p>Proof. To prove the assertion, we apply induction on n. From the inequality</p><p>(2), we observe that l~u1; a2; u26EGu1 a2; u2 EGa1 a2; u2, so the</p><p>130 S. Sezer / Information Sciences 174 (2005) 123142Gj Gj Gj j j j</p><p>j = 1,2, . . .,n. In this case it is easily seen that the maps, dened by</p></li><li><p>S. Sezer / Information Sciences 174 (2005) 123142 131EG1Gnx1; . . . ; xn; y1; . . . ; yn :n^</p><p>j1EGjxj; yj 4</p><p>and</p><p>EG1GnG1Gnx1; . . . ; xn; y1; . . . ; yn; z1; . . . ; zn; t1; . . . ; tn</p><p>:n^</p><p>j1EGjGjxj; yj; zj; tj</p><p>5are fuzzy equalities on G1 Gn and (G1 Gn) (G1 Gn),respectively.</p><p>Proposition 11. For each j = 1,2, . . ., n, let hGj; ~ji be a vague group w.r.t. thefuzzy equalities EGjGj on Gj Gj and EGj on Gj. Let the fuzzy relation ~ :G1 Gn G1 Gn,G1 Gn be defined by</p><p>l~a1; . . . ; an; b1; . . . ; bn; c1; . . . ; cn :n^</p><p>j1l~jaj; bj; cj:</p><p>For the fuzzy equalities EG1Gn and EG1GnG1Gn defined by (4) and(5), respectively, hG1 Gn; ~i is a vague group w.r.t. EG1GnG1Gnand EG1Gn.</p><p>Proof. Let xj,yj 2 Gj and j ord~j for all j = 1,2, . . .,n. Therefore,</p><p>l~x1; . . . ; xn; y1; . . . ; yn; x11y1; . . . ; xnnyn n^</p><p>j1l~jxj; yj; xjjyj 1;</p><p>i.e., the condition (F.1) is satised. We dene</p><p>R : l~a1; . . . ; an; b1; . . . ; bn; c1; . . . ; cn;</p><p>S : l~x1; . . . ; xn; y1; . . . ; yn; z1; . . . ; zn;</p><p>T : EG1GnG1Gna1; . . . ; an; b1; . . . ; bn;x1; . . . ; xn; y1; . . . ; yn:</p><p>Then, for each j 2 J, we have</p><p>R ^ S ^ T 6l~jaj; bj; cj ^ l~jxj; yj; zj ^ EGjGjaj; bj; xj; yj</p><p>6EGjcj; zj:</p></li><li><p>Thus</p><p>R ^ S ^ T 6n^</p><p>j1EGjcj; zj EG1Gnc1; . . . ; cn; z1; . . . ; zn;</p><p>i.e., the condition (F.2) is satised. Thus, ~ is a vague binary operation onG1 Gn.</p><p>Let a = (a1, . . .,an), b = (b1, . . .,bn), c = (c1, . . .,cn), d = (d1, . . .,dn), q = (q1, . . .,qn), m = (m1, . . .,mn), w = (w1, . . .,wn) and</p><p>Y : l~b; c; d ^ l~a; d;m ^ l~a; b; q ^ l~q; c;w:Then, for all j 2 J, we have</p><p>Y 6 l~jbj; cj; dj ^ l~jaj; dj;mj ^ l~jaj; bj; qj ^ l~jqj; cj;wj6EGjmj;wj:</p><p>Therefore, we get</p><p>Y 6n^</p><p>j1EGjmj;wj EG1Gnm1; . . . ;mn; w1; . . . ;wn:</p><p>If ej is the identity element of hGj; ~ji, then</p><p>l~x1; . . . ; xn; e1; . . . ; en; x1; . . . ; xn n^</p><p>j1l~jxj; ej; xj 1</p><p>and</p><p>l~e1; . . . ; en; x1; . . . ; xn; x1; . . . ; xn</p><p>n^</p><p>j1l~iej; xj; xj 1 for all j 1; . . . ; n:</p><p>Hence (e1, . . .,en) is the identity element of hG1 Gn; ~i. On the otherhand, we can write</p><p>l~x1; . . . ; xn; x11 ; . . . ; x1n ; e1; . . . ; en n^</p><p>j1l~jxj; x1j ; ej 1</p><p>and</p><p>l~x11 ; . . . ; x1n ; x1; . . . ; xn; e1; . . . ; en n^</p><p>j1l~jx1j ; xj; ej 1;</p><p>i.e. (x11, . . .,xn</p><p>1) is the inverse of (x1, . . .,x...</p></li></ul>