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Fuzzy Information and Engineering
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ELSEVIER
ORIGINAL ARTICLE
Fuzzy Soft Topological Groups
S. Nazmul • S. K. Samanta
Received: 13 March 2013/ Revised: 29 October 2013/ Accepted: 13 January 2014/
Abstract Notions of Lowen type fuzzy soft topological space are introduced and some of their properties are established in the present paper. Besides this, a combined structure of a fuzzy soft topological space and a fuzzy soft group, which is termed here as fuzzy soft topological group is introduced. Homomorphic images and preimages are also examined. Finally, some definitions and results on fuzzy soft set are studied.
Keywords Soft sets • Fuzzy soft sets • Soft topology • Fuzzy soft topology • Soft topological groups • Fuzzy soft topological groups
© 2014 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.
1. Introduction
In 1965, Zadeh [1] introduced the notion of fuzzy sets and fuzzy set operations. Afterwards attempts have been made to develop several mathematical structures using fuzzy set theory. In 1968, Chang [2] introduced fuzzy topology by axiomatizing some properties of a collection of fuzzy subsets. Subsequently, Lowen [3] slightly changed one of the axioms of fuzzy topology as formulated by Chang, and introduced another definition of fuzzy topology to make it suitable for an idea of good extension of certain properties of topology to fuzzy topology. Rosenfeld [4] formulated the elements of the theory of fuzzy groups and Foster [5] introduced fuzzy topological group.
S. Nazmul (E3)
Department of Mathematics, Govt. College of Education, Burdwan-713102, West Bengal, India email: sk.nazmuL math@yahoo.in S. K. Samanta
Department of Mathematics,Visva-Bharati, P.O.-Santiniketan, Birbhum-731235, West Bengal, India Peer review under responsibility of Fuzzy Information and Engineering Branch of the Operations Research Society of China.
© 2014 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.fiae.2014.06.006
Recently, in 1999, Molodtsov [6] proposed a new approach by introducing 'soft set theory' for modeling vagueness and uncertainties. Subsequently, several mathematical structures have been developed using fuzzy set theory and soft set theory or a combination of these two theories. In 2001-2003, Maji et al. [7, 8] who worked on some mathematical aspects of soft sets and fuzzy soft sets. In 2007, Aktas et al. [9] who introduced a basic version of soft group theory which is further extended to fuzzy soft group by authors [10]. In 2011, Shabir et al. [11] introduced a notion of soft topological spaces. Later Tanay et al. [12] worked on topological structure of fuzzy soft sets and subsequently Simsekler et al. [13], Varol et al. [14] on fuzzy soft topological spaces. As a continuation of this, it is natural to investigate the behaviour of topological structures in fuzzy soft set theoretic form. In 2012, Roy et al. [15] studied some properties of fuzzy soft topology related to fuzzy soft base. In view of this, we have introduced, in this paper, a notion of Lowen type fuzzy soft topology and studied some more properties of this fuzzy topological space. We also introduce fuzzy soft topological group, which is a combined structure of a fuzzy soft topological space and a fuzzy soft group. Homomorphic images and preimages are also examined. Again some definitions and results on fuzzy soft sets are studied.
The organization of the paper is as follows:
Section 2 is the preliminary section where definitions together with some properties of fuzzy soft sets (in our form), fuzzy soft groups, fuzzy topology and soft topology are given. In Section 3, a definition of Lowen type fuzzy soft topology is introduced and some of its properties are studied. In Section 4, a definition of fuzzy soft topological group is presented and some of its properties are examined. In Section 5, the scope of further research towards applications of fuzzy/soft topological group theory are stated. Section 6 is the conclusion section. For the economy of space the straightforward proofs of the theorems are omitted.
2. Preliminary
2.1. Fuzzy Soft Sets
Following Maji et al. [8] some definitions and preliminary results are presented in this section in our form. Unless otherwise stated, X will be assumed to be an initial universal set and A will be taken to be a set of parameters. Let FP(X) denote the set of all fuzzy sets of X and FS (X, A) denote the set of all fuzzy soft sets over X.
Definition 2.1 A pair (F, A) is called a fuzzy soft set over X, where F is a mapping given by F : A —» FP(X). In other words, a fuzzy soft set over U is a parameterized family of fuzzy subsets of the universe X.
Let (Fi,A) and (F2,A) be two fuzzy soft sets over a common universe X. Then {F\,A) is said to be fuzzy soft subset of A) if F\(a) < Fzia), Vn E A. This relation is denoted by (Fi, A)c(F2,A).
Let (Fi,A) and (F2,A) be two fuzzy soft sets over a common universe X. Then (F\, A) is said to be fuzzy soft equal to (F2, A) if(F\, A)C(F2, A) and (F2, A)C(F,, A).
The complement of a fuzzy soft set (F, A), denoted by (Fc, A), is defined by Fc(a) = (F{a))c = complement of the fuzzy subset F(ar), V a e A.
Fuzzy Inf. Eng. (2014)6: 71-92_73
A fuzzy soft set (F, A) over X is said to be a null fuzzy soft set if F (a) = null fuzzy subset of X,4cc<= A, and an absolute fuzzy soft set if F(a) = X, V a 6 A. This is denoted by Ф and A respectively.
Definition 2.2 Let {(F;,A); i e A) be a nonempty family of fuzzy soft sets over a common universe X. Then their
(i) Intersection, denoted by П;€д, is defined by = (fl^F^A), where (П,'едFiXa) = nieA(F;(ar)), V or 6 A.
(ii) Union, denoted by и,ед, is defined by и,-6д(F,-,A) = (игбдР„А), where (U,eAF,)(a) = Uia\(F¡(or)), V or 6 A.
Definition 2.3 Let X and Y be two nonempty sets and f : X —> Y be a mapping. Then
(i) the image of a fuzzy soft set (F, A) 6 FS(X,A) under the mapping f is defined by /(F, A) = (/(F),A), where [f(F)](a) = f[F(a)l V a 6 A;
(ii) the inverse image of a fuzzy soft set (G,A) e S(Y,A) under the mapping f is defined by f-l(G, A) = (f-\G),A), where [/"'(G)] (a) = /"'[С(а)], Чае A.
Proposition 2.1 Let X and Y be two nonempty sets and f : X -» Y be a mapping. If (Fi,A), (F2,A) e FS(X,A), then
(i) (Fi,A)c(F2,A) => /[(Fi,A)]c/[(F2,A)].
(ii) /[(FbA)U(F2,A)] =/[(FbA)]U/[(F2,A)].
(iii) /[(FbA)n(F2,A)]c/[(FbA)]n/[(F2,A)].
(iv) /[(Fi, A)n(F2, A)] = /[(F l, A)]n/[(F2, A)] iff is injective.
Proof Since (FbA), (F2,A) e FS(X,A), it follows that F^or), F2(a) are fuzzy subsets of X, V a e A.
(i) Since (Fb А) с (F2, A), it follows that Fi(a) С F2(a), VaeA
=> flFi(a)] c_/[F2(a)], VaeA =>/[№,A)]c/[(F2,A)].
(ii) For each a e A,
[f(Fi U Fj)](a) = f[Fi(a) иВД] = /[^(or)] U f[F2(a)]
=> №Fi U F2]](ar) = [/(Fi) U f(F2)\(a).
Thus /[(FbA) U (F2, A)] = /[(FbA)] U /[(F2, A)].
(iii) For each a e A,
[/(Fi П F2)](or) = /[Fi(ar) П F2(or)] с f[Fi(a)] П /[F2(a)]
=> [/[F, n F2]](ar) с [/(Fi) П /(F2)](a).
Thus /[(FbA) П (F2, А)] с /[(Fb A)] n /[(F2,A)].
(iv) Follows from part (iii).
Proposition 2.2 Let X and ¥ be two nonempty sets and f : X -* Y be a mapping. If (G1;A), (G2,A) 6 FS(Y,A), then
(i) (G1,A)c(G2,A) => r^Gi.Ayicf-^ifh.A)].
(ii) /_1[(Gi,A)U(G2,A)] = /_1[(Gi,A)]U/_1[(G2,A)].
(iii) /_1[(Gi,A)n(G2,A)] = /_1[(Gi,A)]n/_1[(G2,A)].
Proof Proofs are similar to that of Proposition 2.1.
Proposition 2.3 Let X and ¥ be two nonempty sets and f : X -* Y be a mapping. If (G,A) 6 FS(Y,A), then
(i) flf'1 (G, A)]c(G, A).
(ii) f[f~\G,A)] = (G,A) iffissurjective.
Proof Proofs are similar to that of Proposition 2.1.
Proposition 2.4 Let X and Y be two nonempty sets and f : X —» Y be a mapping. If (F,A) e FS(X,A), then
(i) (G, A)c/_1 [/(G, A)].
(ii) /-'[/(G,A)] = (G, A) ¡// ii injective.
Proof Proofs are similar to that of Proposition 2.1.
Definition 2.4 [4] Let A, B be two fuzzy subsets of X and Y respectively. Then their product denoted by AxB and defined by (AxB)(x,y) = imn{A(x), B(y)}, V (x, y) e Xx Y.
Definition 2.5 [5] The product fxf2: № x X2) -> (Y\ x Y2) of mappings f\ : Xi -» Y\ and f2 : X2 -> Y2 is defined by (/i x f2)(xi, x2) = [f\(x{), f2(x2)] for each (xux2) e № xl2).
Definition 2.6 [5] Let f\ : Xi Yi, f2 : X2 Y2 be two mappings and Ai, A2 be two fuzzy subsets ofYi and Y2 respectively. Then (/1 x ^)_1(Ai x A2) = /f^Ai) x f2\A2).
We now give a definition of cartesian product of two fuzzy sets.
Definition 2.7 Let (F, A) and (G, A) i>e two fuzzy soft sets over X. Then their product is defined as (F, A) x (G, A) = (F xG,A), wftere [FxG](or) = F(a)xG(a), Va e A. /i ii c/ear that (F X G, A) is a fuzzy soft set over XxX.
Proposition 2.5 Let (F, A), (G, A) and (H, A) are fuzzy soft sets over X. Then
(i) (F, A) x [(G, A) U (ii, A)] = [(F, A) x (G, A)] U [(F, A) x (if, A)].
(ii) (F,A) x [(G,A) n (H,A)\ = [(F,A) x (G,A)] n [(F,A) x (H,A)\.
(iii) [(F,,A) x (Gi,A)] n [(F2,A) x (G2, A)]
= [(Fj.A) n (F2,A)] x [(Gi,A) n (G2,A)].
Proof (i) Since (F, A) x [(G, A) U (H, A)] = (F, A) x [(G U H), A]
= [(Fx(GUif)),A].
So for each or 6 A and xi, x2 e X,
{[F x (G U H)\(a)Kxu x2) = [F(a) x [G(a) U H(a)\}(xux2) = [F(a)](xi) A [G(a) U H(a)](x2) = [F(a)](x,) A [[G(or)]te) V [H(a)](x2)} = [[F(«)]Ui) A [G(or)]fe>] V [[F(a)](x,) A [H(a)](x2)] = [[F(a) x G(or)](^i,*2)] V [[F(or) x H(a)](xux2j\ = [[(F X G) U (F X H)\{a)\(xux2).
[F x (G U if)](or) = [(F x G) U (F x H)](a) and hence,
[Hx(GUH),A] = [(F x G) U (F x H), A], Therefore (F, A) x [(G, A) U (if, A)] = [(F, A) x (G, A)] U [(F, A) x (if, A)].
(ii) Proof is similar to part (i).
(iii) Let a e A and xj,x2 e X. Then we have
{[(Fj x Gi) n (F2 x G2)] (ar)K*i, *2) = {[№(«) x Gi(or)] n [F2(or) x G2(o;)]]}(*!, *2) = {[Fi(or) X Gi(a)](j;i, j:2)} A {[F2(a) X G2(a)](x1,x2)} = [F,(«)](*,) A [G,(a)l(x2) A [F2(a)](x,) A [G2(a)](x2) = ([F, (<*)](*,) A [F2(a)](x,)) A {[Gi(a)](x2) A [G2(a)](x2)| = [[Fl(Q-)] n [F2(a)]](x,) A [[G^a)] n [G2(ar)]](*2) = {[[Fi(a)] n {F2(a)J\ x [[G,(gt)] n [G2(qO]])(x,,x2) = {[(Fi n F2) x (Gi n G2)](a)Kx!,x2).
[(Fi x GO n (F2 x G2)](a) = [(Fj n F2) x (Gj n G2)](a) and hence,
[(F1,A)x(Gi,A)]n[(F2,A)x(G2,A)] = [(F1,A)n(F2,A)]x[(G1,A)n(G2,A)].
Definition 2.8 A fuzzy soft set (F, A) over X is said to be constant fuzzy soft set if F(a) = c'a, V a e A, where c^(x) = ca, V * s X and ca e [0,1]. This is denoted by (c,A).
2.2. Fuzzy Soft Groups
Definition 2.9 [10] A fuzzy soft set (F, A) over X is said to be a fuzzy soft group over X iffF(a) is a fuzzy subgroup ofX, V a e A.
Theorem 2.1 [ 10] Lei (Fi, A) and (F2,A)be two fuzzy soft groups over X and (Fi, A) ¿>e a fuzzy soft subgroup o/(F2,A). If f be a homomorphism from X into Y, then
(f(F\),A) and (f(F2),A) are both fuzzy soft subgroups over Y and (f(F{),A) is a fuzzy soft subgroup of(f(F2),A).
Theorem 2.2 [10] Let (Gi,A) and (G2,A) be two fuzzy soft groups over Y and (Gi, A) be a fuzzy soft subgroup of (G2, A). If f be a homomorphism from X to Y, then [/~'(G i),A], and [f-\G2),A] are both fuzzy soft subgroups over X, and [/~'(Gi),A] is a fuzzy soft subgroup of[f~l(G2),A\.
2.3. Fuzzy Topology
Definition 2.10 [3] A fuzzy topology on a set X is a family r of fuzzy sets in X which satisfy the following conditions:
(i) ~c e r, V c e I where c(x) = c, V x e X.
(ii) If A, Bet, then An Bet.
(iii) If Ai 6rVi€ A, then (J A, € r.
The pair (X, t) is called a fuzzy topological space, or ftsfor short. We shall refer to this type of fuzzy topology as Lowen fuzzy topology (briefly LFT) on X.
It is to be noted that in the definition of a fuzzy topology by Chang (which will be refer to by CFT), the condition (i) is replaced by (i') 0, let.
Definition 2.11 [5] Let A be a fuzzy set in X and r, an LFT on X. Then the induced fuzzy topology on A is the family of fuzzy subsets of A which are the intersections with A of t-open fuzzy sets in X. This induced fuzzy topology is called Lowen type subspace fuzzy topology (briefly LSFT) on A and is denoted by ta - The pair (A, ta) is called a fuzzy subspace of (X, r).
If (A, ta), (B, vb) are fuzzy subspaces of fuzzy topological spaces (X, t), (7, v) respectively and if / is a mapping of (X, t) into (Y, v), then we say that / is a mapping of (A, rA) into (B, vB) if /(A) c B.
Definition 2.12 [2,3] Let (X, r) and (y, v) be two fuzzy topological spaces (Chang or Lowen) and f : X —> Y be a mapping. Then f is said to be continuous if f~l(X) 6 r.VAev.
Definition 2.13 [5] Let (A, ta), (B, vg) be fuzzy subspaces of fuzzy topological spaces (X, t), (Y, v) respectively. Then a mapping f of (A, ta) into (B, vg) is said to be relatively fuzzy continuous if for each open fuzzy set V in vb, the intersection /"' [V] fl A is in ta and f is said to be relatively fuzzy open if for each open fuzzy set U in ta, the image /[[/] is in vg.
2.4. Soft Topology
Definition 2.14 [11] Let r be the collection of soft sets over X. Then r is said to be a soft topology on X if
(i) (<f>, A), (X, A) e t where ¡p(a) = <f> and X(a) = X for all a £ A.
(ii) The intersection of any two soft sets in r belongs to r.
Fuzzy Inf. Eng. (2014)6: 71-92_77
(iii) The union of soft sets in r belongs to r.
The triplet (X, a, t) is called a soft topological space over X.
Proposition 2.6 [11] Let (X, A, r) be a soft topological space over X. Then the collection -f = {F(or) | (F, A) e r}/or each a e A, defines a topology on X.
Proposition 2.7 [11] Let (X, A, Ti) and(X,A,T2) be two soft topological spaces over X. Then (X,A,nnr2) where n nr2 = ((F,A) | (F, A) e n & (F,A) e r2} ¡.s a io/i topological space over X. But the union of two soft topological spaces over X may not be a soft topological space over X.
3. Fuzzy Soft Topological Spaces
In this section, following Tanay et al. [12] and Roy et al. [15], we have introduced the definition of Lowen type fuzzy soft topology, a generalized form of enriched fuzzy soft topology as introduced by Varol et al. [14] and studied some of their properties. The development of product fuzzy soft topology is made by approaching in a slightly different manner from that of [14] and some properties are established. Unless otherwise stated, X is an initial universal set; A is the nonempty set of parameters; FS(X,A) denotes the collection of all fuzzy soft sets over X under the parameter set A and P(FS (X, A)) denotes the power set of FS (X, A).
Definition 3.1 Let r be the collection of fuzzy soft sets over X. Then r is said to be a Lowen type fuzzy soft topology on X if
(i) (IT, A) 6 r where ~c(ct) = (Ta, ca £ /, V or 6 A.
(ii) The intersection of any two fuzzy soft sets in r belongs to r.
(iii) The union of fuzzy soft sets in r belongs to r.
The triplet (X, A, t) is called Lowen type fuzzy soft topological space over X. The members of t are said to be Lowen type r-fuzzy soft open sets or simply fuzzy soft open sets in X.
Proposition 3.1 Let (X, A, t) be a Lowen type fuzzy soft topological space over X. Then the collection i" = {F(or) | (F,A) e r) for each a e A, defines a Lowen type fuzzy topology on X.
Proof By definition for any or 6 A, we have r" = (F(or) | (F, A) 6 r}. Now
(i) (c, A) e r implies that c^ e r" for all ca e I.
(ii) Let F and G e r". Then there exists fuzzy soft sets (F,A) and (G, A) e r such that F(a) = F and G(or) = G. Since r is a fuzzy soft topology implies that (FnG, A) e t. Thus (F7)G)(a) = F(a) n G(a) = F n G e -f.
(iii) Let F, e r®, i e A. Then there exists (Fj,A) e r such that F,(a) = F,. Since t is a fuzzy soft topology implies that U1£a(F,-, A) = QJ^Fi, A) e t. Therefore (U,eAF,)(qO = UieAF,(Qr) = U,eAF; e r".
Thus t" defines a Lowen type fuzzy topology on X for each a e A.
Proposition 3.2 If(X,A,T) be a Lowen type fuzzy soft topological space and iff = {(G,A) e FS(X) | G(ar) e r", V a e A), where i", a e A is the topology as defined in Proposition 2.6, then r* is a Lowen type fuzzy soft topology on X such that [t*]« = r", Vor € A.
Proof Since ~c(a) = Fa e r", V a e A => (cj A) 6 r*.
Now let (Fi, A) and (F2, A) e r*. Then Fi(ar), F2(a) er", V a e A.
Fi (or) n F2(qt) e t®, V « e A => (F, nF2)(a) er", V a 6 A.
Thus (Fi, A) n(F2,A) = (Fi nF2, A) e r*. Again let (F„ A) e r*, V i € A. Then F,(a) e r", V i e A, V a e A => (J F,(or) e r", V a e A.
_ i'GA
So ( UieÄF,)(a) er", V a e A. Thus ÜieA(F,-,A) e r*. Therefore r* is a soft topology on X.
Next let [/ 6 Ta. Then 3 (F, A) 6 t such that [/ = F(o-). _ Construct (G, A) e F5 (Z) such that G(a) = F(a) and G()8) = 0, V ß * a. So (G,A) e r* and U = F(a) = G(a) e [r*]a. Therefore
r" c [t']a. (1)
Also let V e [r*]". Then 3 (F, A) 6 r* such that V = F(a) 6 i". Therefore
[t*]" c r". (2)
Thus from (1) and (2), we get r" = [r*]", Vor e A.
Proposition 3.3 Let(X,A,t\) and(X,A, r2) fce two Lowen type fuzzy soft topological spaces over X. Then (X, A, n nr2) where n nr2 = {(F, A) | (F, A) e n & (F, A) e r2} is a Lowen type fuzzy soft topological space over X.
Definition 3.2 Let X and Y be two nonempty sets, r, v be two Lowen type fuzzy soft topologies on X, Y respectively and f : X -» Y be a mapping. The image of r and the preimage ofv under f are denoted by f(j) and f~l(v) respectively, defined by
(i) f(j) = {(G,A) e FS(Y,A) | f~\G,A) = (J-\G),A) er} and
(ii) f~\v) = If~\G,A) = (/-1(G),A) | (G,A) e v).
Proposition 3.4 Let X and Y be two non-empty sets, r, v be two Lowen type fuzzy soft topologies on X, Y respectively and f : X —» Y be a mapping. Then
(i) /~'(v) is a Lowen type fuzzy soft topology on X, and
Fuzzy Inf. Eng. (2014)6: 71-92_79
(ii) /(t) is a Lowen type fuzzy soft topology on ¥.
Proof (i) Since
[LTH^K«)]« = [T'KSX«)]]« = L(c~r)(a)J(/M)
= = ca(y) = ca = Fa(x) = [fö](a)](x), V or e A, VieX.
So (cx,A) = f-\FY,A) e /"», Vc.ei.
Next let (Fi, A), (F2, A) e /"V)-Then 3 (Gi, A), (G2,A) 6 v such that
/_1(Gi,A) = (Fi, A) and/"'(G2,A) = (F2,A). Now (Gi, A), (G2, A) e v => (Gi, A) n (G2, A) e v and
(Fi,A) n (Fi,A) = /-'(Gi,A) n f~\G2,A) = f1[(G,,A) n (G2,A)]. Therefore (Fi,A) n (F2,A) e /"^v). Again let (Fu A) e /_1(v), i e A. Then _3 (G,-,A) 6 v, i6 Asuchthat/_1(G,-,A) = (F,-, A), i e A._ So U,£a(G„A) e v and = U^aLT^A)] = /"'[LU^A)].
Therefore LU^i-A) e f~l(v). Thus f~l(v) is a Lowen type fuzzy soft topology on X.
(ii) From part (i) we have (cx,A) = f'l(cy,A). So by Definition 3.2, we get (cy, A) e f(j).
Next let (Gi, A), (G2,A) 6 /(t).
f~\Gi,A), /~'(G2,A) e r
=> /_1(Gi,A) n_/-1(G2,A) = /"'[(GlA) n (G2, A)] e r. Therefore (Gi,A) n (G2,A) e /(t). Again let (G;, A) e /(t), i e A. Then
f~l(Gi,A) e r, i e A. So Ö,eA/-HG„A) = /-1[UieA(G„A)] e r. Therefore U,eA(G„A) e /(r). Thus /(r) is a Lowen type fuzzy soft topology on Y.
Definition 3.3 Let (F,A) be a fuzzy soft set over X and r be a fuzzy soft topology on X. Then the induced fuzzy soft topology on (F,A) is the family of fuzzy soft subsets of (F, A) which are the intersections with (F,A) of t-open fuzzy saß sets over X. The induced fuzzy soft topology is denoted by t^j [shortly (jp)], and the pair [(F,A),t(f^)] [shortly (F,A,Tp)] is called a fuzzy softsubspace topology of(X,A,r).
Remark 3.1 The induced fuzzy soft topology does not satisfy the condition (i) of Definition 3.1 in general. However, if (F'j,A) e Tf for all j e J, then for each j e J, there exists (Fj,A) e r such that (F'j,A) = (Fj,A) n (F, A).
Now, (F', A) = Üj€j(F'rA) = ÜjeA(Fj,A) n (F, A)] and for each a e A,
[F'(a)](x) = V [*>)](*)
=V [ A w}]
=A{[V[^(a)](x)]' wi«»«}
= [[|jF,(ar)]nF(ar)](;c), VxeX. jeJ
Therefore (F',A) = Ujej(Fj,A) n (F,A) and hence (F',A) e rF.
So, this induced fuzzy soft topology satisfy the conditions (ii) and (iii) of Definition 3.1.
Definition 3.4 Let (X,A,t) and (y,A, v) be two fuzzy soft topological spaces. A mapping f of (X, A, t) into (Y, A, v) is said to be fuzzy soft continuous if for each open fuzzy soft set (G,A) in v the inverse image /"' [(G, A)] = [/"'(G), A] is in r and f is said to be fuzzy soft open if for each open fuzzy set (F,A) in r, the image f[(F,A)] = [/(F), A] is in v.
If (F, A, tf) and (G, A, vg) are fuzzy soft subspaces of fuzzy soft topological spaces (X, A, t), (F, A, v) respectively and if / is a mapping of (X, A, r) into (Y, A, v), then we say that / is a mapping of (F, A, i>) into (G, A, vg) if /[(F, A)] c (G, A).
Definition 3.5 Let (F, A,tp) and (G, A, vq) be fuzzy soft subspace of fuzzy soft topological spaces (X,A,t) and (Y,A, v) respectively. Then the mapping f of (F, A, 1>) into (G, A, vg), ii said to be relatively fuzzy soft continuous iff for each fuzzy soft open set (V, A) in vg, the intersection /"' [(V, A)] n (F, A) is in Tp and f is said to be relatively fuzzy soft open iff for each fuzzy soft open set (U,A) in tf, the image f[(U, A)] is in vg-
Proposition 3.5 If (F, A, Tp) and (G,A,vg) are fuzzy soft subspaces of fuzzy soft topological spaces (X,A,t), (Y,A,v) respectively and f is a fuzzy soft continuous mapping of(X,A,r) into (Y,A,v) such that /[(F,A)] c (G,A). Then f is relatively fuzzy soft continuous mapping of(F, A, tp) into (G, A, vq).
Proof Let(V',A) e vG. Then there exists (V, A) e v such that (V, A) = (V,A)n(G,A) and/^KKA)] 6i. Hence
r\(V',A)] n (F,A) = A)] n /_1[(G,A)] n (F,A)
= /~1[(V,A)] n (F,A) e rF (since (F, A) c /_1 [(G, A)]).
So / is relatively fuzzy soft continuous mapping of (F, A,tf) into (G, A, vg).
Definition 3.6 A bijective mapping f : (X,A,t) —> (F.A, v) is said io be fuzzy soft homeomorphism if f is fuzzy soft continuous and fuzzy soft open. A bijective map-
Fuzzy Inf. Eng. (2014)6: 71-92_81
ping f : (F,A,tf) -* (G,A, vg) is said to be relatively fuzzy soft homeomorphism if f[(F, A)] — (G, A) and f is relatively fuzzy soft continuous and relatively fuzzy soft open.
Proposition 3.6 Let f : (X, A, r) —> (Y,A, v) and g : (Y, A, v) -»(Z, A, oj) be two fuzzy soft continuous (or fuzzy soft open) mappings. Then the composition g.f : (X, A, r) —> (Z, A, a>) is a fuzzy soft continuous (or fuzzy soft open) mapping.
Proof Let Off, A) e a>. Since g : (Y,A,v) —> (Z,A,co) is a fuzzy soft continuous mapping, it follows that g~l [(H, A)] e v.
Again since / : (X,A, r) —> (Y,A, v) be a fuzzy soft continuous mapping, it follows that/_l[g_,[(/i,A)]] e r
So the composition g • f : (X, A, t) —> (Z, A, eo) is a fuzzy soft continuous mapping.
The proof in the case of relatively fuzzy soft open mapping is similar.
Proposition 3.7 Let (F,A,Tp), (G,A,va)and(H,A,ajH)befuzzysoftsubspacesof fuzzy soft topological spaces (X,A,t), (Y,A,v) and (Z,A,cS) respectively. Let f : (F,A,Tp) —* (G,A,vg), g : (G,A,vq) —> (H,A, wj) be two relatively fuzzy soft continuous (or relatively fuzzy soft open) mappings. If f[(F, A)] c (G,A), then the composition g ■ f : (F,A,Tp) —> (H,A, u>H) is a relatively fuzzy soft continuous (or relatively fuzzy soft open) mapping.
Proof Let (W,A) e Since g : (G, A, vg) -» (//, A, is a relatively fuzzy soft continuous mapping, it follows that g'1 [(W, A)] n (G, A) e vq.
Again since / : (F, A, tf) -»(G, A, vg) be a relatively fuzzy soft continuous mapping, it follows that
rV [W A)] n (G, A)] n (F, A) etj
=» (g' /)"'P,A)] n /"'[(G,A)] n (F,A) = (g ■ /r'[(W,A)] n (F.A) e rF (since (F,A) c /_1[(G,A)]). Thus the composition g ■ f : (F, A, t>) -* (H, A, <dh) is a relatively fuzzy soft continuous mapping.
The proof in the case of relatively fuzzy soft open mapping is trivial.
Definition 3.7 Let r be a fuzzy soft topology on X. A subfamily /3 ofT is said to be a base for t if each member of t can be expressed as a union of members of p.
Definition 3.8 Let r be a fuzzy soft topology on X and the induced fuzzy soft topology on (F, A). A subfamily /3' o/tf is said to be a base for tf if each member of tp can be represented as a union of members of ft.
Proposition 3.8 Let f : (X, A, r) -» (Y,A, v) be a mapping and ¡3 be a base for v. Then f is fuzzy soft continuous iff for each (G, A) in /3 the inverse image /_1[(G,A)] is in t.
Proof The only if part is obvious.
Suppose the given condition is satisfied. Let (H,A) 6 v. Then there exists (Hi, A) e P, i g A such that (H, A) = u,eA(ff„ A) and f1 [(ff„ A)] er.ieA.
Hence f~l\(H,A)\ = f-l[UieA(H„A)i = U,^[f-l[(H„A)\\ e r. So / is fuzzy soft continuous.
Proposition 3.9 Let (F,A,Tjr), (G,A,va)be fuzzy soft subspaces of fuzzy soft topological spaces (X,A,t), (7,a, v) respectively. Letfi' be a base for va- Then a mapping f : (F,A,tp) —> (G,A,vq) is relatively fuzzy soft continuous iff for each (V,A) in /?' the intersection /"' [(V, A)] n (F, A) is in Tp.
Proof The proof is similar to that of Proposition 3.8.
Proposition 3.10 Let (X,A,t) and (Y,A,v) be two Lowen type fuzzy soft topological spaces. Then T = {(F,A) x (G,A) | (F,A) e r, (G, A) e v} forms an open base for a fuzzy soft topology onXxY.
Proof We note that (c^y, A) = (£*, A) x (cj, A).
Since (cx,A) 6 r and (cy, A) e v, follows that (¿xjy, A) e IT. Again let (F!,A) x (Gi,A), (F2,A) x (G2iA) where (Fi,A), (F2,A) e t and (Gi,A), (G2,A) 6 vbe any two members of T. So (Fi,A) n (F2,A) e r and (Gi, A) n (G2, A) e v. Thus
[(FlA) x (Gi,A)] n [(F2,A) x (G2, A)]
= [(F1;A) n (F2,A)] x [(Gi,A) n (G2,A)] e r. Therefore ^ forms an open base for a fuzzy soft topology on X x K
Definition 3.9 The fuzzy soft topology in XxY induced by the open base T is said to be the product fuzzy soft topology of the fuzzy soft topologies r and v. It is denoted by t xv. The fuzzy soft topological spaces [X x Y, A, rxv] is said to be the fuzzy soft topological product of the fuzzy soft topologies (X, A, t) and (Y, A, v).
Proposition 3.11 Let {(Xj,A,Tj), j- 1,2} be two fuzzy soft topological spaces and (X,A,t), the product fuzzy soft topological space. For each j = 1,2, let (Fj, A) be a fuzzy soft set over Xj and (F, A) be the product fuzzy soft set over X. Then the induced fuzzy soft topology Tf on (F, A) has, as a base, the set of product fuzzy soft sets of the
form U(U'rA), where (U),A) e (r>y, j = 1,2. Proof From Proposition 3.10, r has a base 2
p = {[~[(t/;, A) I (Uj,A) e tj, j = 1,2}.
So the base for Tp is given by
Pf = {( n (F, A) | (Uj,A) e tj, j = 1,2}
= { f] \(Uj,A) n (F;,A)] I (Uj,A) e t„ j = 1,2)
= { I WpA) = (Uj,A) n (Fj,A) e (r>l
This product fuzzy soft subspace is denoted by (F,A,1>) = ni^i. A, (t, )/■,.].
Proposition 3.12 Lei (X, A, r) and (y, A,v)be two fuzzy soft topological spaces. Then the projection mappings Jtx '■ (X x Y, A, txv) -» (X, A, r) and ny : (XxY,A, txv) —> (y, A, v) are fuzzy soft continuous and fuzzy soft open. Also txv is the smallest fuzzy soft topology in X x Yfor which the projection mappings are fuzzy soft continuous.
Proof (F,A) e r => n^1 [(F, A)J = (F,A) x (ly,A) is a basic open fuzzy soft set in
So nx is fuzzy soft continuous.
Again let (G, A) e rxv. Then there exists a sub family T' = {[(t/„ A), (V„A)], j e A} of T such that (G,A) = U,eA[((/„A) x (Vj,A)]. Let (U,A) = U(£/;, A) and (V,A) = U(Vi, A). Then, for each or e A and x e X, we have
[MG)](tf)]« = MG(ar)]](*)
^[IJ^xViKor)]]«
teA i€A
= v [U[c/i(oi)x v'(ar)]](x'>')
(xjOerr-'GO teA
= V yft^WWAlVi^O')]
(x,y)arx-Hx) ¡«A
= V [ V [[^WIW A WWKjo]]
feA (x^orx-'W
= V [[£/,(<*)]« A [ Y rKMlOO]]
ieA fcwOeni-'GO
= V A [ V [^(a)](y)]](since(x,y) e n~J{x), VyeY)
ieA yeY
= \J[lUi(a№)Aci,a].
So, kx[(G,A)] = Uiea[(t/;, A) n (ci, A)J e r since (i/„A) n (ct,A) e t. Therefore jtx is fuzzy soft open.
Similarly, it can be shown that ny is also fuzzy soft continuous and fuzzy soft open. Next let co be any fuzzy soft topology on X x Y such that the mappings nx : (X x Y,A,cn) -> (X,A,f) and nY \ (Xx Y,A,w) -> (Y,A,v) are fuzzy soft continuous.
Let [(i/,A)x(V, A)] be any basic open fuzzy soft set in rxv. Now
[(17, A) x (V,A)] = [(17, A) n (h?,A)] x [(l~y,A) n (V, A)] = [(17, A) x (h, A)] n [(û, A) x (V, A)] = nj1 [(Ï7, A)] n ny1 [(V, A)] e oj (since if^[(U,A)\, 7Ty [(V,A)J e cj).
Thus t x v subset of co.
Therefore r x v is the smallest fuzzy soft topology in Xx Y for which the projection mappings are fuzzy soft continuous.
Proposition 3.13 Let (X, A, r) be the product space of two fuzzy soft topological spaces (Xi,A,n) and (Xi,A,T2) and tt, : (X,A,t) —> (Xj,A,Tj), i = 1,2 be the projection mappings. If(Y,A, v) be any fuzzy soft topological space, then the mapping f : (y,A, v) —» (X,A,t) is fuzzy soft continuous iff the mappings itif : (Y,A,v) —» (Xi,A,Ti), i= 1,2 are fuzzy soft continuous.
Proof First, let / : (F, A, v) -» (X, A, t) be fuzzy soft continuous.
Also Jti : (X, A, t) —> (Xi, A, r,) is fuzzy soft continuous, i = 1,2. Then by Proposition 3.6, Tiif is fuzzy soft continuous, i = 1,2.
Conversely, let tt,/ is fuzzy soft continuous, i = 1,2.
Let (F,A) 6 r and T = {(U,A) x (V,A) \ U,A) e n, (V,A) e r2}. Then 3 a sub family T' = {(i/„A), (Vj,A) \ i e A} of <T such that (F,A) = U,[(i7„A) x (V„A)]. Thus
r1 [(F, A)] = r1 [UiCCC/i, A) x (Vi, A)]]
= r1[Û/[jrr1(^.A)n^1(V,.A)]] = Ui[(^1/)-1([/i,A) n (K2f)~\Vi,A)] 6 v.
Therefore / is fuzzy soft continuous.
Following [14], the Proposition 3.14 and Proposition 3.15 are presented in this section in our form which will be used in studying fuzzy soft topological group in the next section.
Proposition 3.14 Let (X,A,r) be the product space of two fuzzy soft topological spaces (Xi,A,Ti) and (X2,A,t2) and (Y,A,v) be the product space of two fuzzy soft topological spaces (Yi,A,v{) and (Y2,A,v2). If the mappings fj of (Xj,A,Tj) into (Yj,A,Vj), j = 1,2 are fuzzy soft continuous, then the product mapping f = f\Xf2 of (X,A,t) into (y,A, v), defined by f(x\, x2) = (fi(xi),f2(x2)) is fuzzy soft continuous.
Fuzzy Inf. Eng. (2014)6: 71-92_85
Proof Since[nYif\{xux2) = JtYllMxi), f2(x2)] = /i(x,) = f\l7rx,,x2)J = [ftftxA (xi,x2) implies that jtYlf = /i^ • Also f\ and nxx are fuzzy soft continuous and hence from Proposition 3.6, Try,/ is fuzzy soft continuous.
Similarly, ?ry2/ is fuzzy soft continuous. Therefore from Proposition 3.13, / is fuzzy soft continuous.
Proposition 3.15 Let (X,A, r) be the product space of two fuzzy soft topological spaces (X\,A,T\) and (X2,A,t2) and (Y,A, v) be the product space of two fuzzy soft topological spaces (Yt,A, vi) and (Y2,A,v2). If the mappings fj of (Xj,A,Tj) into (Yj,A,vf), j = 1,2 is fuzzy soft open, then the product mapping f = /i x f2 of (X,A,t) into (F,A, v), defined by fix i,x2) = (/i(xi),/2(x2)) is fuzzy soft open.
Proof Let (U,A) g r. Then there exist open fuzzy soft sets (Ujm,A) e tj, j = 1,2, m e A such that (U,A) = UmeA[([/lm,A) x (i/2m,A)l. Now,
[[/(£/)](«)]№ =XT[[/l(tflm)](<*) X [/2(C/2m)](a)](y)
= V V [Lt/lmf^ja,)] x [[i72w(«)Jfe)] = V V V [[tfi-(«)Ka>] A [^2-(«)]Ca)], y = &i>3'2)
mEA ziE^'Cyi) aeJf'Oi)
= V {[ V [^l-C^Kzi)] A [ V [%.(«*)]<»)]}
= V {[/iftWaOHO'i) A f/2[i/2m(a)110v2)} me A
=XJ[[/i(t/im)](a)] X L/2(i/2m)](or)J(j).
Thus/[(If, A)] = U^L/K^A] x [/2(£/2m),A]].
Since fj, j =1,2 are fuzzy soft open, /[(£/, A)] is open in v and hence the product mapping / = /i x/2 of (X,A, r) into (K, A, v), defined by /(x,, x2) = (/i(xi),/2(x2)) is fuzzy soft open.
Proposition 3.16 Let (X,A,t) be the product space of two fuzzy soft topological spaces (Xi,A,Ti) and (X2,A,t2). Let (Fi,A), (F2, A) be fuzzy soft sets over Xi, X2 respectively and (F, A) be the corresponding product fuzzy soft set. If(Y, A, v) be any fuzzy soft topological space, (G, A) be a fuzzy soft set over Y and f be a mapping of (G, A, vg) into (F, A, tf), then f is relatively fuzzy soft continuous iff7tjf is relatively fuzzy soft continuous for each j = 1,2.
Proof Let / be relatively fuzzy soft continuous. Also since nj, j = 1,2 are fuzzy soft continuous and hence by Proposition 3.5, Kj, j = 1,2 are relatively fuzzy soft continuous. So by Proposition 3.7, Ttjf, j =1,2 are relatively fuzzy soft continuous. Conversely, let Ttjf, j =1,2 be relatively fuzzy soft continuous. Let (H,A) = (HuA)x(H2,A), where (HUA) e t1fi and (H2,A) e t2,2. Then by
Proposition 3.10, the set of such (H,A) forms abase for tf. Since
/-'[(Я,А)] П (0,А) = Г1Щ\НиА) П^ЧНг.А)] n (G,A)
= [(Я1fT\HuA] n (G,A)] П [Ш)~\Н2,А\ n (G,A)]
is open in vc, as Ttjf, j = 1,2 are relatively fuzzy soft continuous. Therefore by Proposition 3.9, / is relatively fuzzy soft continuous.
Proposition 3.17 Let(X,A,r), (Y, A, v) be the product spaces of (Xi, A,n), (X2,A,t2) andof(YuA,vi), (Y2, A, v2) respectively. Let(F,A) = (FuA)x(F2,A), where (FUA), (F2,A) are fuzzy soft sets over Xi, X2 respectively and (G,A) = (Gi,A)x(G2,A), where (Gi,A), (G2,A) are fuzzy soft sets over Y], Y2 respectively. If the mappings fjOf(Fj,A,Tjr) into (Gj,A,VjG), j = 1,2 are relatively fuzzy soft continuous (open), then the product mapping f = fi X-fi of(F, A, Tf) into (G, A, vg), cloned by f(x\ ,x2) = (fi(xi),f2(x2)) is relatively fuzzy soft continuous (open).
Proof Proof follows from Proposition 3.14 and Proposition 3.15.
Proposition 3.18 Let (X, А, т) be a fuzzy soft topological space. Then the mapping f : (X, А, т) —»(X, A, t) defined by f(x) = x, VxeX is fuzzy soft continuous.
Proof Let (F, A) e r. Then /-1 [(F, A)] = [/"' (F), A] = (F, A) б т. Therefore / : (X, А, т) —> (X, А, т) is fuzzy soft continuous.
Proposition 3.19 Suppose (X, A, r) and (У, A,v)be two fuzzy soft topological spaces. Then the mapping f : (X, А,т) —» (Y,A, v) defined by f(x) = yo, VieX, where yo is a fixed element of Y is fuzzy soft continuous.
Proof Let (F, A) 6 v. Then
(IT1 (F)](a)K*) = {Г'^ШО) = [f («)](/«}
= №)](yo) = caC«ry)V;teX.
So [/_1(^)](а) = ^ and hence /_1 (F, A) = [j'-\F),A\ = (с, A) e r.
Therefore the mapping / : (Х,А,т) —» (У,А,у) defined by f(x) = yo, V x e X, where yo is a fixed element of У is fuzzy soft continuous.
Proposition 3.20 Let (Х,А,т) be the product space of two fuzzy soft topological spaces (Xi,A,ri) and (X2,A,t2). Let a e Xi. Then the mapping f : (X2,A,t2) —» (X, A, t) defined by f(x2) = (a, x2) is fuzzy soft continuous.
Proof Let я, : (X, A, r) -»(Xit A, r,-), i= 1,2 be the projection mappings.
Now n\f : (X2,А,т2) —» (Х\,А,Т]) is such that n\f(x2) = a, V x2 e X2 and n2f : (X2, А, т2) -> (X2, А, т2) is such that n2f(x2) = x2, Vx2 e X2.
So by Propositions 3.18 and 3.19, mappings яif and n2f are fuzzy soft continuous. Therefore by Proposition 3.13, / is a fuzzy soft continuous.
Proposition 3.21 Let (X, А, т) be the product space of two fuzzy soft topological spaces (Xi,A,ti) and (X2,A,t2). Let (F\,A), (F2,A) be two fuzzy soft sets over
Xi, Xi respectively and (F,A) be the corresponding product fuzzy soft set over X. Then for each a e Xi such that [Fi(ar)](a) > [F2(a)](x2), V x2 g X2 and V a g A, the mapping f : (F2,A,t2fi) —> (F,A,tf) defined by f(x2) = (a,x2) is relatively fuzzy soft continuous.
Proof Since
{[/№)](<*)}(«, *2) = lF2(a)](x2)
= {[F,(ar)](a) A [F2(a)](x2)} = [F(a)]{a,x2), V g X2, V a e A.
Also if Xi + a, then
{[f(F2)i(a))(xux2)= 0
<{[/■,(<*)!(*,) A [F2(ar)](*2)} = [F{aj\{xl,x2), V xi g Xu V x2 e X2, V a g A.
Therefore /[(F2,A)] c (F,A) and hence by Proposition 3.20 and Proposition 3.5, the mapping / : (F2,A,t2f2) -» (F,A,tf) defined by f(x2) = (a, x2) is relatively fuzzy soft continuous.
Definition 3.10 A mapping f : (X,A, t) —> (7, A, v) is said to be fuzzy soft homeo-morphism iff is bijective and /, /"' are fuzzy soft continuous.
4. Fuzzy Soft Topological Group
Let X be a group. Suppose (G, A) be a fuzzy soft group and r be a fuzzy soft topology over X. Let / : (X x X,A,tXt) -> (X,A,t) and g : (X,A,r) -» (X,A,t) be two mappings defined by f(x, y) = xy and g{x) = x'1. Since
{[/(GxG)](a)}(x) = \/ {[GxG](q-)}(zi,z2) fa^/"1« = \f {[G(a)](zi) A [G(a)]fe)}
fa ¿2 lef'W
< [G(a)](ziz2) [since G(a) is a fuzzy group]
= [G(a)](x), V x g X and V a g A.
Then /[(GxG,A)]c(G,A).
{[g(G)](a)}W= \/ [G(a)](z)
= \J [G(œ)](z_1) [since G(a) is a fuzzy group]
= [G(or)](jc), V xeX and V ar e A.
Theng[(G,A)] = (G,A). _
Hence / : (GxG,A,tgxtg) -> (G, A, rG) and g : (G,A,tg) -> (G,A,tg) are two mappings.
Next note that (G, A, tg) is a fuzzy soft subspace of (X, A, r) and (G, A, tg)x(G, A, to) a fuzzy soft subspace of the product fuzzy soft space (X, A, t)x(X, A, t).
Definition 4.1 Lei X be a group and r a Juzzy soft topology over X. Let (G, A) be a fuzzy soft group over X. Then (G,A,tg) is said to be fuzzy soft topological group over X if the mappings
(i) / : (G,A,tg)x(G,A,tg) -» (G,A,tg), defined by f(x,y) = xy, and
(ii) g : (G, A, To) —> (G,A,tg), defined by g(x) = x'1, where x,y e X are relatively fuzzy soft continuous.
Proposition 4.1 Let X be a group with fuzzy soft topology r and (G, A) be a fuzzy soft group over X. Then (G, A, tg) is a fuzzy soft topological group over X iff the mapping h : (G,A,tg)x(G,A,tg) —> (G, A, tg), defined by f(x,y) = xy'1 is relatively fuzzy soft continuous.
Proof Let (G,A,to) be a fuzzy soft topological group. By Proposition 3.17, the mapping! : (G,A,tg)x(G,A,tg) -> (G, A,tg)x(G,A,tg) defined by i(x, y) = (x,y_1) is relatively fuzzy soft continuous.
Also since (G, A, tq) be a fuzzy soft topological group, it follows that / : (G,A,tg)X(G,A,Tg) (G,A,Tg), defined by /(x, y) = xy is relatively fuzzy soft continuous.
Hence by Proposition 3.14, the mapping h = fi: (G,A,tg)x(G,A,tg) —> (G,A, tg), defined by fi(x,y) = f(x,y~l) = xy'1 is relatively fuzzy soft continuous.
Conversely, let the mapping h : (G,A,tg)x(G,A,tg) -» (G,A,tg), defined by h(x,y) = xy'1, be relatively fuzzy soft continuous. If e be the identity element of X, then [G(<z)](e) > [G(or)](^), V x e X and V a e A. So by Proposition 3.21, the mapping j : (G,A,tq) -» (G,A,to)x(G,A,tg), defined by j(y) = (e,y) is relatively fuzzy soft continuous. Thus the mapping g = hj : (G,A,tg) -» (G,A,tg), defined by hj(y) = h(e,y) = ey'1 = y'1 is relatively fuzzy soft continuous.
Again since k: (G, A, to) —> (G, A, to), defined by k(x) = x is relatively fuzzy soft continuous and hence by Proposition 3.17, the mapping i : (G,A,tg)x(G,A,to) —» (G,A,to)x(G,A,to), defined by i(x,y) = (x, is relatively fuzzy soft continuous.
Fuzzy Inf. Eng. (2014)6: 71-92_89
Thus f = hi: (G,A,tg)x(G,A,tg) -> (G, A,tg), defined by hi(x, y) = h(x,y~l) = x[y~,]~l = xy is relatively fuzzy soft continuous.
Therefore (G, A, tg) is a fuzzy soft topological group.
Proposition 4.2 Let e be the identity element of a group X having fuzzy soft topology т and (G, A, to) be a fuzzy soft topological group over X. Then for each a e Ge = [x e X | [G(or)](x) = [G(a)](e), Чае A), the mappings Ra : (G, A, rc) -> (G, A, rc), La : (G, A, tg) —» (G, A, tg), defined by Ra(x) = xa and La(x) — ax, are relatively fuzzy soft homeomorphisms.
Proof Since
{[^(G)](Qr)}« = [G(ar)](jra_1) > min{\G(a)](x), [G(<*)](a)J = {[G(a)](x) Л [G(a)](e)} = [G(a)](x) = [G(a)](xa^a) > min{[G(a)](xa"'), [G(a)](a)} = ([G(a)](xa_1) A [G(a)](e)} = [G(a)](xa"1) = {Ra[G(a)W(x), V x e X and V a e A,
it follows that Да[(G, A)] = (G,A).
Similarly, La[(G,A)] = (G,A), V a € Ge.
Also L„ = fi where i: у -»(a, y) and / : (je, y) —> xy.
Since [G(a)](a) > [G(a)](y), V у e X, it follows from Proposition 3.21, i is relatively fuzzy soft continuous mapping of (G,A,to) into (G,A,tg)x(G,A,tg) and / is relatively fuzzy soft continuous by hypothesis. Hence La is relatively fuzzy soft continuous and therefore L"1 = L„-i also continuous. Hence La is relatively fuzzy soft homeomorphism. Similarly, we can prove that Ra is relatively fuzzy soft homeomorphism.
Proposition 4.3 Let X, ¥ be two groups, v be a fuzzy soft topology over Y and т = f'l(v). If f be a homomorphism of (X, А, т) into (Y,A, v) and (G,A, vc) is a fuzzy soft topological group over ¥, then [/_1(G),A,Ty- 1(g)] is a fuzzy soft topological group over X.
Proof Sincer = f~l(v), the mapping / : (Х,А,т) -» (У, A, v) is fuzzy soft continuous. Further (G, A, vG) is a fuzzy soft topological group over Y.
So, (G,A) is a fuzzy soft group over Y and the mapping hy : (у\,уг) -» of (G, A, vg)x(G, A, vg) into (G, A, vg) is relatively fuzzy soft continuous.
Now (f~' (G),A,Tf-i(Cjj) is a fuzzy soft subspace of (Х,А,т) and by Proposition 3.17, the product mapping /х/ of [/_1(G),A,t/-i(g)]x[/-1(G), A,Tf-l(0)\ into (G, A, vg)x(G, A, vg) is relatively fuzzy soft continuous.
Again / is a homomorphism and hence by Theorem 2.2, [f~1 (G), A] is a fuzzy soft group over X.
Lethx : (x,,x2) -» xix2' be a mapping of [/-1(G),A,t/-i(g)]x[/-1(G),A,t/-i(c)] into [f-'(G),A,Tf-i(G)] and (U,A) e rf-i(G).
Since / is a fuzzy soft continuous mapping of (X,A,t) into (7, A, v), by Proposition 3.5, / is a relatively fuzzy soft continuous mapping of [f~\G),A,rf-i(p)] into (G, A, vg). Further since r = /_1 (v), there exists a fuzzy soft set (V, A) e vg such that f~lV(V,A)] = (U,A).
Since Iiy is relatively fuzzy soft continuous, it follows that
[hy1 (V, A) n [(G, A) x (G, A)]] e [vGxvG]. Also since / x / is relatively fuzzy soft continuous, it follows that (f x frl[h?(Y,A) n [(G, A) x (G,A)]] n [[/_1(G),A] x [/_1(G),AJ] = (/ x /r'^'CV.A)] n [[/"'(G), A] x [/-'(G),A]] € [rr,(G) x Ty-.^]-
{[/^(£/)](a)Kx,,x2) = [U(a)][hx(xux2)] = [U(a)^x~2l)
= {r\V(amxix-2l) = \V(a)][f(xlX-21)]
= [V(a)]mxi) (/fe))"1] = {hyl[V(a)]}mx i),/fe)]
= {(/x/)-1[/ip1[V(a)]]}(x1,x2), V(ji,x2)eXxX, VoreA.
Thus ft"1 [(i/,A)] = (/x/T'^KKA)]] and hence hx'[(U, A)] n [f~](G)xf~](G)] e
Tf-l(G)XTf-<(G).
So, hx is relatively fuzzy soft continuous and therefore [/_1(G),A,ry-i(G)] is a fuzzy soft topological group over X.
Proposition 4.4 Let X, Y be two groups, r be a fuzzy soft topology over X and v = /(t). If f be a one-one homomorphism of (X, A, r) into (Y, A, v) and (F, A,tf) is a fuzzy soft topological group over X, then [/(F), A, v/^)] is a fuzzy soft topological group over Y.
Proof Since (F, A, rF) is a fuzzy soft topological group over X, it follows that (F, A) is a fuzzy soft group over Zand the mapping hx : (x\,x2) -» XiX^1 of (F, A, i>)x(F, A, TF) into (F, A, tf) is relatively fuzzy soft continuous.
Also since / is soft open and hence by Proposition 3.16, the product mapping fxf of (F, A, tf) x(F,A,tf) into [/(F), A,vf(fj] x [/(F), A, v/m] is relatively fuzzy soft open.
Again / is a homomorphism and hence by Theorem 2.1, [/(F), A] is a fuzzy soft group over Y.
Let hY : (yuy2) -» yiy2l be a mapping of [f(F),A,vm]x[f(F),A,vf^] into [/(F), A, vm] and (V, A) 6 vm. Then 3 (U,A) e v such that (V, A) = (U, A) n (F, A). Therefore, /_1[(t/,A)] e t and since / is one-one, it follows that /_1[(K A)] = /"' [(i/, A)] n (F, A) e tf.
Again since hx is relatively fuzzy soft continuous, we have hx LT1 l(y, A)]] n [(F, A) x (F, A)] e [tf x tf].
K/X/TW№)]])(*!,*2) = [V(<*)]]№l),/(*2>] = [W№i)(/fer']
= [(№x1r1])[V(a)]](x1,x2), V jci, x2eX, V a e A.
So (/x/r^KKA)]] = ih-'f-'m^A)].
Since / is one-one, / is open and hence by Proposition 3.17, / x / is relatively fuzzy soft open.
Again since / x / is relatively fuzzy soft open, it follows that (/x/^'LTHKA)] n [(F,A)x(F,A)]] e [v/(F)xv/(f)] => (/ x/)[(/x/r'^-'WA)] n [(F,A) x (F,A)]] 6 [v/m x vf(F)] => hY'[(V,A)] n [/(F,A) n /(F,A)] e [v/(F) x vf(F)]. Therefore hy is relatively fuzzy soft continuous and hence [/(F), A, v/y?)] is a fuzzy soft topological group over Y.
5. Scope of Further Research towards Applications
Topological groups have many applications in the field of abstract integration theory such as the development of the theory of Haar measure and Haar integral and also in manifold theory through the development of Lie groups. In this paper, we have introduced a more generalized structure of topological groups which is called fuzzy soft topological group. Here we have integrated 'Fuzziness' and 'Softness' which is best suited for modeling physical systems involving uncertainty. In this sense, this study has a great significance. In this paper, we have just introduced the notion of topological groups in fuzzy soft setting and studied some of their basic properties. There is an ample scope of further research in developing the theory of Haar measure; Haar integral and Lie group theory in this setting to reach the applicational frontier of the fuzzy/soft topological group theory.
6. Conclusion
There is a scope for studying the fuzzy soft topological group structure with soft topologies in the sense of Hazra, Majumdar and Samanta [16] and fuzzy soft groups with varying parameter sets. This problem will be studied in our subsequent papers.
Acknowledgements
The authors express their sincere thanks to the reviewers and the editors for then-valuable suggestions in writing the paper in the present form. The present work is partially supported by Special Assistance Programme (SAP) of UGC, New Delhi, India [Grant No. F 510/8/DRS/2009 (SAP -II)].
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