Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring

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1 Advances in Fuzzy Mathematics. ISSN X Volume 12, Number 2 (2017), pp Research India Publications Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring K. Meena BS & H Department, Muthoot Institute of Technology and Science, Varikoli, India. Abstract In this paper some properties of intuitionistic fuzzy ideals of an intuitionistic fuzzy ring is discussed. The notion of characteristic intuitionistic fuzzy subring of an intuitionistic fuzzy ring is introduced and proved that it is an int. fuzzy ideal. It s characterization in terms of level sets is provided. Moreover some lattices and sublattices of intuitionistic fuzzy subrings and intuitionistic fuzzy ideals of a given int. fuzzy ring are constructed. Also lattices of characteristic intuitionistic fuzzy subrings possessing sup-property and its sublattices are constructed. Keywords: Intuitionistic Fuzzy Subring, Intuitionistic Fuzzy Ideals, Characteristic Int. Fuzzy Subrings, Complete Lattices, Generated Int. Fuzzy Subring 2010 Mathematics Subject Classification: 13A15 1 INTRODUCTION The idea of fuzzy sets introduced by L.A. Zadeh (1965) [27] is an approach to mathematical representation of vagueness in everyday curriculum. In 1971, A. Rosenfeld [24] initiated the study of applying the notion of fuzzy sets in group theory. N. Ajmal and K. V. Thomas [3], [4] studied the lattice structure of fuzzy algebraic structures and also proved its modularity. The concept of a normal fuzzy subgroup of fuzzy group was introduced by Wu [26]. Besides this, Martinez [19] studied the properties of fuzzy subring of a fuzzy ring. N Ajmal and I. Jahan [5] investigated the properties of fuzzy sets of fuzzy group and studied the lattice

2 230 K. Meena structure of fuzzy subgroups of a fuzzy group. In 1983 K. T. Atanassov [9] introduced the notion of intuitionistic fuzzy sets, which is a generalization of fuzzy sets. The foundation laid by K. T. Atanassov, to the introduction of intuitionistic fuzzy sets, has tremendously inspired the development of intuitionistic fuzzy abstract algebra, which has been growing actively since then. The idea of intuitionistic fuzzy subgroup initiated by R. Biswas in [12] illustrates the flourishment of Intuitionistic fuzzy sets in a more generalized way. Likewise in [11] Banerjee and Basnet introduced Intuitionistic fuzzy subrings and ideals. Many researchers have applied the notion of intuitionistic fuzzy sets to the fields of Sociometry, Medical diagnosis. Decision Making, Logic Programming, Artificial Intelligence etc. [1, 2, 10, 14, 17, 28]. In this paper the lattice structure of characteristic intuitionistic fuzzy subrings of intuitionistic fuzzy ring in a commutative ring is studied. The remainder of the paper is organized as follows: In section 2 and 3 some definitions and results of Intuitionistic fuzzy sets and intuitionistic fuzzy ideals of int. fuzzy ring are reviewed. In section 4, characteristic intuitionistic fuzzy subset of intuitionistic fuzzy ring and its basic properties are introduced. Its characterisation in terms of level sets is provided. It is proved that the inf-supstar family of characteristic intuitionistic fuzzy sets form a sublattice of the lattice of intuitionistic fuzzy ideals of int. fuzzy ring. In section 5 and 6 the lattice structure of characteristic intuitionistic fuzzy subrings with sup-property is studied. Moreover various sublattices of intuitionistic fuzzy subrings are investigated. 2 PRELIMINARIES In this section some basic concepts applied in this paper are recalled [20 23]. Let (R,+, ) be a commutative ring. Definition 2.1.Let X be a non-empty set. An intuitionistic fuzzy set in X (IFS(X)) is defined as an object of the form A = { x, µa(x),νa(x) /x X} where µa : X [0,1] and νa: X [0,1], define the degree of membership and the degree of non-membership for every x X. Definition 2.2.Let A = { x, µa(x),νa(x) /x X}, B = { x, µb(x),νb(x) /x X} be two IFS(X). Then (i) A B iff for all x X,µA(x) µb(x) and νa(x) νb(x). (ii) A B = { x,(µa µb)(x),(νa νb)(x) /x X}

3 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 231 (iii) A B = { x,(µa µb)(x),(νa νb)(x) /x X} (iv) A ᵒB = { x,(µa ᵒµB)(x),(νAᵒνB)(x) /x X} where (µa ᵒµB)(x) = {µa(y) µb(z)/y,z X, yz = x} and (νaᵒνb)(x) = {νa(y) νb(z)/y,z X, yz = x}. Definition 2.3.Let {Ai}i I be an arbitrary family of IFS(X) where Ai = { x, µai(x),νai(x) /x X} i I, then (i) Ai = { x, µai(x), νai(x) /x X} (ii) Ai = { x, µai(x), νai(x) /x X}. The set of all int. fuzzy sets in X (IFS(X)) constitutes a complete lattice under the ordering of intuitionistic fuzzy set inclusion,, with µa(x) = sup[µa(x)] and νa(x) = inf[νa(x)]. The notions of level subset At and strong level subset A > t are defined as follows: For A IFS(X), t [0, 1] (i) At = {x X: (µa) (x) t, νa(x) t} (ii) A > t = {x X: µa(x) > t, νa(x) < t}. Definition2.4. Let A = { x, µa(x),νa(x) /x X} be an IFS(X).Then is called the tip of A. Definition 2.5.An intuitionistic fuzzy subset A = { x, µa(x),νa(x) /x R} of R is said to be an intuitionistic fuzzy subring of R (IFSR(R)) if for all x,y R, (i) µa(x + y) µa(x) µa(y) (ii) µa(xy) µa(x) µa(y) (iii) νa(x + y) νa(x) νa(y) (iv) νa(xy) νa(x) νa(y). Definition 2.6. Let A IFS(R). Then A is said to have the sup-property if for each non empty subset Y of R there exists a y 0 Y such that

4 232 K. Meena sup yєy µa(y) = µa(y 0 ) and inf yєy νa(y) = νa(y 0 ). In view of the fact that arbitrary intersection of IFSR(R) is an IFSR(R), the following definition of intuitionistic fuzzy subring generated by an intuitionistic fuzzy set laid the foundation for the study of lattice theoretic aspect of intuitionistic fuzzy algebraic structures. Definition 2.7.[20] Let A = { x, µa(x),νa(x) /x R} be an IFS(R). Then the int. fuzzy subring generated by A is defined to be the least int. fuzzy subring of R denoted as A and defined as A = { x, µa, νa /x R} where µa = {µ: µa µ, µ IFSR(R)} and νa = {ν: νa ν, ν IFSR(R)} Definition 2.8.Let A = { x, µa(x),νa(x) /x R} be an IFSR(R). Then A is called an int. fuzzy ideal of R (IFI(R)) if, for all x, y R, (i) µa(x y) µa(x) µa(y) (ii) µa(xy) µa(x) (iii) νa(x y) νa(x) νa(y) (iv) νa(xy) νa(x). Let f: X Y and A IFS(X), B IFS(Y). Then the int. image f (A) = { y, f(µa)(y),f(νa)(y) /y Y } is defined as. The int. inverse image of Y, f 1 (B) = {x, f 1 (µb)(x), f 1 (νb)(x) /x X} is defined as f 1 (µb)(x) = µb(f(x)), f 1 (νb)(x) = νb(f(x)). Proposition 2.9.[7] Let A,B IFS(R). Then (ii) (A ᵒB) t = At Bt, t [0, 1] provided A, B possess sup-property. Proposition [7] Let {Ai} IFS(R). Then

5 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 233 (i) ( Ai) > t = (Ai) > t, t [0,1] (ii) ( Ai)t= (Ai)t, t [0,1]. Proposition 2.11.[23] Let A IFS(R) with tip. Then the following are equivalent (i) A IFSR(R) (ii) At is a subring of R, t [0,1] > (iii) A t is a subring of R, t [0, 1]. Proposition 2.12.[7] Let A IFS(R) with tip t0 = µa(0). Then A t > = A t> t [0, µa (0)]. Proposition 2.13.[23] Let A IFSR(R). Then the following are equivalent: (i) A IFI(R) (ii) At is an ideal of R, t [0, µa(0)] (iii) A t > is an ideal of R, t [0, µa (0)]. 3 INT. FUZZY IDEALS OF AN INT. FUZZY RING In this section, the notion of an int. fuzzy ideal of int. fuzzy ring is introduced and its related properties are studied. Some characterizations of the notion of int. fuzzy ideal of int. fuzzy ring is discussed. Int. fuzzy analogues of certain results of classical ring theory is obtained. Definition 3.1. Let A, B IFS(R). Then A is said to be an int. fuzzy subset of B (IFS(B)) if A B. Lemma 3.2.Let A = { x, µa(x),νa(x)) /x R} and B = { x, µb(x),νb(x) /x R}be IFS(R). Then (i) A B iff. At Bt t [0,1] (ii) A B iff. A > t Bt > t [0, 1]. Proof. (i) Let A B then µa(x) µb(x),νa(x) νb(x). Let x (µa)t µa(x) t then clearly x (µb)t. Also if x (νa)t νa(x) t then clearly x (νb)t. Conversely if At Bt, t [0, 1] then A B, t [0, 1] (ii) It follows directly from (i).

6 234 K. Meena Definition 3.3. Let A IFS (B). Then the int. fuzzy subring of B generated by A is the least IFSR (B) containing A defined as follows: μ A B= {µ IFSR (B): µa µ} γ A B= {ν IFSR (B): νa ν}. Definition 3.4. Let A, B IFSR(R). Then A is said to be intuitionistic fuzzy subring of B [IFSR (B)] if A IFS (B). Definition 3.5.Let A IFSR (B). Then A is said to be an IFI (B) if for all x, y, R µa(x + y) µa(x) µa(y) µa(xy) µb(x) µa(y) νa(x + y) νa(x) νa(y) νa(xy) νb(x) νa(y). Theorem 3.7.Let A IFS (B) with tip t0. Then the following are equivalent: (i) A IFI(B) (ii) At is an ideal of Bt, t [0,1] (iii) A > t is an ideal of B t >, t [0,1] (iv) A > t is an ideal of Bt >, t ImA (v) At is an ideal of Bt, t ImA [t ImB: t t0]. Proof. i ii Let A IFI (B). Let x,y At µa(x) t, νa(x) t and µa(y) t, νa(y) t. Let y Bt µb (y ) t, νb(y ) t. Then, µa(x + y) min(µa(x),µa(y)) t, and µa(xy ) min(µa(x),µb(y )) t νa(x + y) max(νa(x),νa(y)) t, νa(xy ) max(νa(x), νb(y )) t. Hence At is an ideal of Bt t [0, 1]. ii iii Let At be an ideal of Bt, t [0,t0]. Now A > t = r>t A r and B t > = r>t Br. Clearly A > t is an ideal of B t >.

7 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 235 iii iv Obvious. iv i Let A > t be an ideal of Bt >, t ImA. Suppose A IFI (B) then for x, y R, µa(x + y) < µa(x) µa(y), µa (xy) < µb(x) µa(y) and νa(x + y) >νa(x) νa(y), νa(xy) >νb(x) νa(y). Let t 1 = µa(x + y) implies x + y μ At1 and t 1 Im µa. Also t 1 < µa(x) µa(y) µa(x) >t 1, µa(y) > t 1 > x, y μ At 1 > But x + y μ At1 is a contradiction. If t = μ A (xy) t <µ B (x) µa(y) x μ > > B,y μ t A and t ImµA. t > But xy μ At is a contradiction. Similarly, if t 1 = νa(x + y) implies x + y γ A t 1 and t 1 ImνA. Also t 1 >νa(x) νa(y) νa(x) < t 1, νa(y) < t 1 > x, y μ At 1 But x + y ν > At. This is a contradiction. 1 If t2 = νa (xy) then t2 >νa(x) νb(y) νa (x) < t2, νb(y) < t2. x ν > > At, y ν 2 Bt and t2 ImνA. 2 > But xy ν At is a contradiction. 2 Since A > t is an ideal of Bt > t Im A, the above is a contradiction and hence A IFI(B). i v Obvious v i Let At be an ideal of Bt t Im A [t ImB: t t0]. Let x, r R. Take t = µa(x), t 1 = µb(r).

8 236 K. Meena Case i: Let t t 1. This implies that t0 µa(x) t 1 = µb(r). Thus t 1 Im µb and t 1 t0. Since At is an ideal of B t 1, xr μ A t 1, i.e. µa (xr) t 1 µa(x) µb(r) Also let t = µa(x), t 1 = µa(y), x,y R. Then, t0 t = µa(x) µa(y) = t 1 and t t0. Therefore x µat, x µat1 and y µat1 µa(x + y) t 1 = µa(y) µa(x) µa(x + y) µa(y) µa(x). Similarly it follows clearly that for x, y R, νa(x + y) νa(x) νa(y) and νa(xy) νb(y) νa(x). Case ii: Let t t 1. This implies that t0 µa(x) = t µb(r). It follows that µb(r) µa(x) = t. Thus t Im µb, x µat and r µbt. Hence xr µat µa(xr) t = µa(x) µb(r). Therefore µa(xr) µa(x) µb(r). Also let t = µa(x), t 1 = µa(y), x,y R. Then t0 µa(x) = t µa(y) = t 1 t = µa(x) µa(y) = t 1. It follows that x µat,y µat and hence x + y µat. i.e., µa(x + y) t = µa(x) µa(y). Therefore µa(x + y) µa(x) µa(y). Similarly for x, y R, νa(x + y) νa(x) νa(y), νa(xy) νa(x) νb(y). It follows that A IFI (B). Theorem 3.8.Let A = { x, µa(x),νa(x) /x R} be an int. fuzzy ideal of B. Then A IFI (B). Proof. Let t0 = µa(0) and t [0,1]. Then by theorem 3.7 At > is an ideal of Bt > t [0,t0]. By proposition 2.12 A > t = At >. This implies that At >is an ideal of B t >. Hence by theorem 3.7 A IFI (B). Proposition 3.9.Let {Ai} IFI (B) be any family. Then (i) i Ai IFI (B).

9 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 237 (ii) i Ai IFI(B). Proof. (i) Let t0 = sup {Ai} = sup { Ai}. Then t0 = sup {supai}. Also Ai IFS (B). Let t [0,t0]. Then by theorem 3.7 (Ai) > t of the family {(Ai) > t} is an ideal of Bt >. By proposition 2.10 i (Ai) > t = ( i Ai) > t. The clearly ( i Ai) > t is an ideal of Bt >. Hence by theorem 3.7, i Ai IFI (B). (ii) Let t0 = sup { i Ai}. Then t0 = inf{sup Ai}. Also i Ai IFS (B). Let t [0,t0]. Since Ai IFI(B), and by theorem3.7 for each i, (A i ) > t is an ideal of Bt >. By proposition 2.10 i (A i ) t > = ( i A i ) t > i Then clearly ( i A i ) t > is an ideal of B t >. Hence by theorem 3.7 i Ai IFI (B). 4 CHARACTERISTIC INTUITIONISTIC FUZZY SUBRING OF INTUITIONISTIC FUZZY SUBRING In [13, 15, 16, 25, 29] the researchers have extended the concept of characteristic subgroup in fuzzy setting. In [18] characteristic subgroup of a fuzzy group was studied. Here in this section the notion of a characteristic intuitionistic fuzzy subring of an int. fuzzy ring is being introduced. Its characterization in terms of level subsets is discussed. Moreover it is proved that the inf-sup star family of characteristic intuitionistic fuzzy set of an int. fuzzy subring is a complete sublattice of the lattice of intuitionistic fuzzy ideals of an int. fuzzy ring. Definition 4.1. Let Q IFS (B) with tip t0. Then Q is said to be a characteristic intuitionistic fuzzy subset of B if µq(tx) µq(x), T A(µB)t, t [0,t0] νq(tx) νq(x), T A(νB)t, t [0,t0], where A(µB)t, A(νB)t is the group of automorphisms of (µb)t and (νb)t. The set of characteristic intuitionistic fuzzy subsets of B is denoted by CIFS (B). Remark 1.B is a characteristic intuitionistic fuzzy subset of B itself. Theorem4.2.Let Q IFS (B) with tip t0. Then Q CIFS (B)) iff Qt is a characteristic subset of Bt t [0,t0].

10 238 K. Meena Proof. Necessary Part: Let t [0,t0] and Q CIFS(B). Then for x (µb)t Similarly for x (νb)t, µq(tx) µq(x),t A(µB)t, t [0,t0] µq(tx) t, if x (µq)t, T A(µB)t Tx (µq)t, if x (µq)t, T A(µB)t T (µq)t (µq)t, T A(µB)t. νq(tx) νq(x),t A(νB)t, t [0,t0] νq(tx) t, if x (νq)t, T A(νB)t Tx (νq)t, if x (νq)t, T A(νB)t T (νq) t (νq) t, T A (νb) t. Qt is a characteristic subset of Bt for each t [0,t0]. Sufficient Part follows by reversing the arguments. Let t [0,t0] and Qt be a characteristic subset of Bt. Then T(µQ)t (µq)t, T A(µB)t for x (µq)t, Tx (µq)t, T A(µB)t µq(tx) t, t [0,t0],T A(µB)t µq(tx) µq(x) if x (µq)t,t A(µB)t µq(tx) µq(x) T A(µB)t, t [0,t0]. Similarly since Qt is a characteristic subset of Bt T(νQ)t (νq)t, T A(νB)t for x (νq)t, Tx (νq)t T A(νQ)t νq(tx) t, t [0,t0],T A(νB)t νq(tx) νq(x) if x (νq)t,t A(νB)t νq(tx) νq(x) T A(νB)t, t [0,t0]. Hence the result. The set of int. fuzzy sets of A possessing sup-property is denoted by IFSs (A). Theorem 4.3. Let A, B IFSs (A). Then A B and A B IFSs (A). Proof. Clearly follows.

11 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 239 The following result gives a generalisation of sup-property. Proposition 4.4.Let A = { x, µa(x),νa(x) /x X} IFS(X), where X is a non-empty set. Then A possesses sup-property iff each non-empty subset B of ImA is closed under arbitrary supremum and infimum, i.e., if sup μ B = b0 then b0 B and if infγ B = b1 then b1 B. Definition 4.5.A non-empty subset Y of the unit interval [0, 1] is said to be a infsupstar subset if every non-empty subset A of Y is closed under arbitrary supremum and infimum i.e., if supµa = a0 then a0 A and if infνa= b0 then b0 A. Remark Every subset of a inf-supstar subset is a inf-supstar subset. 2. Let A IFS(R) then A possesses sup-property iff ImA is a inf-supstar subset. Definition 4.6.Let {Ai}i I IFS(R). Then {Ai}i Iis said to be inf-supstar family if i ImAi is a inf-supstar subset. Proposition 4.7.Let {A i } iєi IFS (B) be an inf-supstar family. Then i ) Ai IFSs(B) for each i I. (ii) i Ω A i ЄIFS s (B)whereΩ I. Proof. (i) Given {A i } iєi is a inf-supstar family. Then i ImgA i is a inf-supstar subset. To prove that Ai IFSs(B), it suffices to show that Im Ai is a inf-supstar subset for each i I. But i ImAi is a inf-superstar subset. Hence every subset of i ImgA i is closed under arbitrary infimum and supremum. Therefore Ai IFSs (B) for all i I. (i) Given {A i } iєi is a inf-supstar family. Then i ImgA i is a inf-supstar subset i ImµAi is closed under arbitrary unions. Let Ω I. Then for x R Im i Ω µai = x R {( i Ω µai) (x)} = x R {sup iєω (µai(x))} = x R μ Ax where μ Ax = {µai(x)/i Ω}. Then µ Ax = {µai(x)/i Ω} ImµAi. But every subset of iєi Imgμ Ai is closed under arbitrary unions. Hence sup µ Ai µ Ax iєi Imgμ Ai Thus Im i I µ Ai i I Imµ Ai. (1)

12 240 K. Meena Similarly given {Ai}i I is a inf-supstar family. Then iєi ImA i is a inf-supstar subset i I ImνAi is closed under arbitrary intersection. Let Ω I. Then for x R, Im i Ω ν Ai = x R {( i Ω ν Ai (x)} = x R {inf(ν Ai (x)/i Ω)} = x R {infν Ax } where γ Ax = {νai(x)/i Ω}. Then γ Ax = {νai(x) /i Ω} i I ImνAi. Since every subset of Im γ Ai is closed under arbitrary intersection inf ν Ax ν Ax i I Imν Ai Thus Im i Ω ν Ai i I Imν Ai (2) Hence from (1) and (2) iєω Ai IFSs (B). It follows from the above result that each member of a inf-supstar family possesses sup-property. In particular, for A, B IFS(R) if ImA ImB is a inf-supstar subset then A and B are said to be jointly inf-supstar. Proposition 4.8.Let Ai = {<x, µai(x),νai(x)>/x R} IFS(R), i I, I = {1, 2... n}. Then {Ai} is a inf-supstar family iff Ai satisfies sup-property for each i I. Proof. Let{A i }, I= {1,..., n} be a inf-sup star family. Then i I Im Ai is a inf-supstar subset which implies that i I Ai has sup-property. Hence AI has sup-property for all i I. Conversely let Ai satisfy sup-property for all i I. Then Ai IFSs(R) Im i I Ai is a inf-supstar subset i I Ai is a inf-supstar subset {A i } iєi is a inf-supstar family. Proposition 4.9.If {Ai} IFS(R) is a inf-sup-star family then ( Ai) t = (Ai) t t [0, 1]. Proof. Given{A i } iєi is a inf-sup-star family implies ImAi is a inf-supstar subset. Also

13 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 241 ( Ai) t = {x R/ µai(x) t, νai(x) t} and (Ai) t= {x R/µAi(x) t, νai(x) t} For x ( Ai)t µai(x) t, νa i(x) t. Let iєi µ Ai = t o and iєi Ai = t 1 then t0,t1 ImAi where t0 t and t1 t. Hence x (Ai)t ( Ai)t (Ai)t (3) Conversely let x (Ai)t. Then x (μ A1 )t (μ A2 )t... and x (γ A1 )t (γ A2 )t... such that (μ A1 (x) μ A2 (x)...) t and (γ A1 (x) γ A2 (x)...) t. Therefore x ( µ Ai ) t and x ( Ai ) t. Hence x ( Ai) t. Therefore Then from (3) and (4) ( Ai) t = (Ai) t. x (Ai)t ( Ai)t. (4) Theorem 4.10.If {Ai} IFS (B) is a maximal inf-supstar family then {A i } iєi is a complete lattice under the order of fuzzy set inclusion. Proof. Let{A i } iєi IFS (B) be a maximal inf-supstar family. Then i I Im Ai is a inf-supstar subset i I Imµ Ai, i I Imν Ai are inf-supstar subsets. Now to show that{a i } iєi is closed under arbitrary supremum and infimum. Let Ω I. To show that i Ω µ Ai {Ai} and i Ω ν Ai {Ai}. Since {Ai} is a infsupstar family Im i Ω µ Ai i I Im µ Ai,Hence Im i Ω µ Ai [ i I Im µ Ai ]= i I Im µ Ai, (5) Also i I Im µ Ai,is a inf-supstar subset. Hence Im i Ω µ Ai [ i I Im µ Ai ] is a inf-supstar subset. Thus i Ω µ Ai U {µ Ai : i I} is a inf-supstar family. By the maximality of inf-supstar family, i Ω µ Ai {µ Ai : i I} = {µ Ai : i I}. Therefore, i Ω µ Ai {µ Ai : i I}. Similarly, since {A i } iєi is a inf-supstar family it implies that Im i I υ Ai i I Im υ Ai,. Hence

14 242 K. Meena (Im i I υ Ai ) [ i I Im υ Ai,] = i I Im υ Ai, (6) Clearly Im νai is a inf-supstar subset. Hence (Im i I υ Ai ) [ i I Im υ Ai,] is inf-supstar subset. Thus i Ω υ Ai U {υ Ai : i I} is a inf-supstar family. By the maximality of inf-supstar family (7) i Ω υ Ai {υ Ai : i I} = {υ Ai : i I} Therefore i Ω υ Ai {υ Ai : i I}. It follows that {Ai}i I is a complete lattice under the ordering of fuzzy set inclusion. Theorem Let {A i } iєi CIFS (B). Then (i) Ai CIFS (B). (ii) Ai CIFS (B) provided {Ai} is a inf-supstar family. Proof.(i) Given {A i } iєi CIFS(B) µ Ai (Tx) µ Ai (x), T A(µB)t t [0,t0] and υ Ai (Tx) υ Ai (x), T A(νB)t, t [0,t0] where t0 is the tip of Ai. Clearly, Ai = {<x, µ Ai υ Ai >/ x R} and i I A i IFS(B). Let t0 = sup ( i µ Ai ) and t [0,t0]. Then t0 = inf sup (µ Ai ) so that t t0 sup µai i I. Let x (µb) t, T A(µB)t, t [0,t0] i I µai(tx)=inf µ Ai (Tx) inf µai(x) = i I µai(x) (8) Similarly let i I υ Ai IFS (B). Let t0 = inf( i υ Ai ) and t [0.t0]. Then t0 = sup inf(υ Ai ) so that t inf υ Ai t0 for each i I. Now for x (νb) t, T A (νb)t, i I υ Ai (Tx) = sup υ Ai (Tx) sup υ Ai (x) (9) = i I υ Ai (x).

15 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 243 From (8) and (9) i I Ai CIFS (B) (ii) For i I, Ai = {<x, µ Ai, υ Ai >/x R}. Clearly i I µ Ai IFS (B) and let t0 = sup{ µ Ai }. Then t0 = sup [sup µ Ai ]. Let t [0,t0]. Given {Ai} CIFS (B) {µ Ai }t,, {υ Ai }t is characteristic subset of (µb)t and(νb)t. Given {Ai} IFS (B) is a inf-supstar family. ( Ai) t = (Ai)t i.e., ( µ Ai ) t = (µ Ai ) t and ( υ Ai ) t = (υ Ai ) t Since arbitrary union and intersection of characteristic subsets of a ring R is a characteristic subset, ( µ Ai )t and ( υ Ai ) t are characteristics subsets of Bt. By theorem 4.2. i I A i CIFS (B). Theorem Let A CIFS (B). Then A IFI (B). Proof. Let x, r R. Take t o = µb(x) µa(r). t o supµ A, x (µ B ) to, r (µ A ) to. Define Then T r A [(µb)t0]. T r : (μ B ) to (μ B ) t o є Tr(x) = xr. Since A CIFS (B) (μ A ) t o is a characteristic subset of (μ B ) t o Tr(μ A ) t o (μ A ) t o Tr A(μ B ) t o Since x (μ B ) t o, Tr(x) (μ A ) t o µa(tr(x)) t0 = µb(x) µa(r) µa(xr) µb(x) µa(r). Let x, y R. Choose t1 = µa(x) µa(y) t1 sup µa, x (µa)t1, y (µa)t1. Define Ty: (μ B ) t 1 (μ B ) t 1 є Ty(xy) = x + y. Then Ty (μ B ) t 1. Since A CIFS (B) (μ A ) t 1 is a characteristic subset of (μ B ) t 1 Ty(μ A ) t 1 (μ A ) t 1, Ty A(μ B ) t 1. Since x (μ A ) t 1, Ty x (μ A ) t 1. µa(t y x) t1 = µa(x) µa(y) µa(x + y) µa(x) µa(y).

16 244 K. Meena For x, r R let t2 = νb(x) νa(r) x (ν B ) t2 r (ν A ) t2,t2 inf νa. Define Tr: (γ B ) t 2 (γ B ) t 2 є Tr(x) = xr. Then Tr A(ν B ) t2. Since A CIFS (B), (ν A ) t2 is a characteristic subset of (ν B ) t2 Tr (ν A ) t2 (ν A ) t2 Tr A(ν B ) t2 Since x (ν B ) t2, Tr(x) (ν A ) t2 νa(tr(x)) t2 = νb(x) νa(r). νa(xr) νb(x) νa(r). Similarly, let x, y R and t3 = νa(x) νa(y) x (ν A ) t3 y (γ B ) t 3 and t3 inf νa. Define Ty: (γ B ) t 3 (γ B ) t 3 є Ty(x) = x + y. Clearly Ty A(γ B ) t 3. Since A CIFS (B) implies (γ A ) t 2 is a characteristic subset of (γ B ) t 3. Ty (γ B ) t 3 (γ B ) t 3, Ty A(γ B ) t 2. Since x (γ A ) t 2, Ty(x) (γ A ) t 3 Hence A IFI (B). νa(ty(x)) t3 = νa(x) νa(y) νa(x + y) νa(x) νa(y). Remark 3. By above result CIFS (B) IFI (B) IFS (B). By proposition 3.9, IFI (B) is closed under arbitrary unions and intersections. Hence forms a complete sublattice of IFS(B). From Theorem 4.11 CIFS (B) is closed under arbitrary intersections. Hence CIFS (B) is a lower complete lattice and is a complete lattice. If CIFS (B) and IFI(B) are inf-supstar families then CIFS(B) IFI(B) IFSs(B) (By theorem 4.12 and Proposition 4.7). Also CIFS (B) is closed under arbitrary unions (By theorem 4.11) with least element identically zero function. So CIFS (B) is an upper complete sublattice of IFI(B). Also IFI(B) is upper complete sublattice of IFSs(B). Theorem 4.13.Let A CIFS (B) with tip t0. Then T (µa/(µb)t) = µa/(µb)t, T A(µB)t, t [0,t0] and T(νA/(νB)t) = νa/(νb)t T A(νB)t, t [0,t0]. Proof. Let t [0,t0] and T A(µB)t. For y (µa) t T (µa/(µb)t)(y) = sup Tx=y [µa/(µb)t(x)] = sup [µa(x)] Tx=y

17 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 245 = µa(t 1 (y)) ( T A(µB)t) µa(y) ( A CIFS(B)) = µa/(µ B ) t (10) Similarly, for y (µ B ) t [µa/ (µb)t](y) = µa(y) = µa(tt 1 (y)) [ T A(µB)t] µa(t 1 (y)) [ A CIFS(B)] = (T 1 ) 1 µa(y)) [By definition of pre-image] = T(µA/(µB)t)(y). (11) From (10) and (11) T(µA/(µB)t) = µa/(µb)t. Also for t [0,t0] and T A[(νB)t]. Then for y (νb)t, T (νa/ (νb)t)(y) = inf Tx=y (νa/(νb)t(x)) =inf Tx=y( ν A (x) = νa(t 1 (y)) ( T A[(νB)t]) νa(y) ( A CIFS(B)) = νa/(νb)t. (12) Similarly for y (νb) t [νa/(νb)t](y) = νa(y) = νa [TT 1 (y)] νa [T 1 (y)] ( A CIFS(B)) =γ A (T 1 ) 1 (y) = T(νA/(νB)t). (13) Hence by (12) and (13) T(νA/(νB)t) = (νa/(νb)t). Definition 4.14.Let A IFSR (B). Then A = {<x, µa(x),νa(x)>/x B} is said to be a characteristic intuitionistic fuzzy subring of B(CIFSR(B)) if A CIFS(B). Remark 4. Clearly an int. fuzzy ring is a characteristic int. fuzzy subring of itself.

18 246 K. Meena Theorem 4.15.Let A IFSR (B). Then A CIFSR (B) iff At is a characteristic subring of Bt, t [0,t0]. Proof. Clearly follows. Example. Consider the ring R = (Z 4, +, ) where Z 4 = {0, 1, 2, 3} and let (2 n ) be the integral multiples of 2 where n is a fixed natural number. Let A = {<x, µa(x), νa(x) >/x Z 4 } be an IFS(Z 4 ) defined as Let B = {<x, µb(x), νb(x)>/x Z 4 } be an IFS (Z 4 ) defined as Then A and B are IFSR (Z 4 ) and B A. For t [0, 1], the level subring Bt is a characteristic subring of At. Hence B CIFSR(A). Theorem 4.16.Let A, B CIFS (P) be jointly inf-supstar. Then A B CIFS (P). Proof. Let t0 = sup(aob)(t) and t [0,t0]. Since A,B are jointly inf-supstar, then ImA ImB is an inf-supstar subset A,B possess sup-property (by proposition 4.8). Hence (A ob)t = At Bt (By proposition 2.9). Let r, s be the tips of A and B, then t0 = min{r, s} so that t r, t s. Given A, B CIFS (P) implies {(µa)t, (νa)t}, {(µb)t,(νb)t} are characteristic subsets of {(µp)t,(νp)t}. Hence At Bt is a characteristic subset of Pt (A ob)t is a characteristic subset of Pt Ao B CIFS(P). The following results are straightforward. Theorem 4.17.Let A CIFSR (B). Then A IFI (B). Theorem 4.18.Let A, B CIFSR (P) be jointly inf-supstar. Then AoB CIFSR (P).

19 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 247 Theorem 4.19.Let A CIFSR (B) with tip t0. Then (i) T(µA/(µB)t) = µa/(µb)t T A(µB)t, t [0,t0]. (ii) T(νA/(νB)t) = νa/(νb)t T A(νB)t, t [0,t0]. Theorem 4.20.Let A CIFSR (B) and D CIFSR (A). Then D CIFSR (B). Proof. Given A = {<x,µa(x),νa(x) >: x B} and D = {<x,µd(x),νd(x) >: x B} Also D CIFSR(A) Dt is a characteristic subring of At t [0,t0] where t0 is the tip of D. Also, t µd (0) µa (0), and t νd(0) νa(0) and A CIFSR(B). Hence Dt is a characteristic subring of Bt. Consequently D CIFSR (B). Theorem 4.21.Let A IFI (P) and B CIFSR (A). Then B IFI (P). Proof. Given B CIFSR (A) Bt is a characteristic subring of the subring At. Also since A IFI (P), At is an ideal of Pt. Hence Bt is a characteristic subring of an ideal At of a ring Pt B IFI (P). 5 INT. FUZZY SUBRINGS AND SUP-PROPERTY The notion of sup-property introduced by Rosenfeld [24] finds prominence in all fields of fuzzy algebraic structure. Ajmal [6] constructed new lattices of fuzzy normal subgroups, possessing sup-property. In this section it is proved that the int. fuzzy subring generated by characteristic int.fuzzy subset possessing sup-property is a characteristic int.fuzzy subring of the parent int.fuzzy ring. Theorem 5.1.[8] Let A IFSR(B). Define an int. fuzzy set Aˆ = {<x, µˆa(x), νˆa(x)>/x R}, where µˆa(x) = sup{r : x <(µa)r>} and νˆa(x) = inf{r : x <(νa)r>}. Then Aˆ IFSR (B) and Aˆ = <A>. Theorem5.2. [8] Let A IFSR (B) and possesses sup-property. Then <A> possesses sup-property. Theorem 5.3.Let A CIFS (B) and possesses sup-property. Then <A> CIFSR (B) having sup-property. Proof. Suppose<A>is not an CIFSR (B) <A> CIFSR (B) <A> = {<x, <µa>, <νa>>} CIFSR (B). Then there exists t o [0, μ B (0)] < μ A t 0 > is not a characteristic subring of (μ B ) to T o A(μ B ) to є T o (< μ A > to ) < μ A > to Hence there exists a y0 R y0 <µa>t0 but y0 T0(< μ A >) to. Thus

20 248 K. Meena y0 =T0(x0) where x0 < μ A > to <µa>(x0) t o sup{r : x0 <(µa)r>} t0. r Imµa Since A possesses sup-property, r0 Im μ A є x o <(μ A ) r o > and r0 t0 (14) As x o < (μ A ) ro > we get x0 = a0a1...an where ai (μ A ) ro Hence T0x o = T 0 (a0a1...an) = T 0 a0 T 0 a1... T 0 an. Also as µa CIFS (B), µa (T 0 ai) µa (ai) and since ai (μ A ) r0 T 0 a i (μ A ) r0. Consequently T 0 x o < (μ A ) ro >. But y0 = T0(x0). Hence y0 < (μ A ) ro > µa(y0) r0 t0 which is a contradiction. Similarly, if <A> is not an CIFSR (B) <A> CIFSR (B) t0 [0, γ B (0)] < υ A > t0 is not a characteristic subring of (υ B ) t0. T o A(γ B ) to є T o (< γ A > to ) < γ A > to Hence there exists y0 R y0 < γ A > to but y0 T0 (< γ A > to ) Thus y0 = T0(x0) where x0 < γ A > to <γ A >(x0) t0. Since A possesses sup-property r0 Im γ A x0 <νa>r0 and r0 t0. As x0 <νa>r0 we get x0= b0b1b2...bn, where bi (γ A ) ro. Hence y0 = T 0 x0 = T0(b0b1b2...bn) = T0(b0)T0(b2)...T0(bn) Also as A CIFS(B) implies νa(t0bi) νa(bi) and since bi (γ A ) ro it follows that T0bi (γ A ) ro. Hence y o =T 0 x o < γ A > ro <γ A >(y o ) r o t o

21 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 249 which is a contradiction. Therefore <A> CIFSR (B) and moreover <A> possesses the sup-property. Hence the result. 6 LATTICES AND INTUITIONISTIC FUZZY IDEALS In this section it is proved that the set of int. fuzzy ideals and the set of characteristic int. fuzzy subrings which possesses sup-property constitutes a sublattice of the lattice of int. fuzzy subrings of a given int. fuzzy ring. Theorem 6.1.The set IFI (B) is a complete lattice of the set of IFSR (B). Proof. Let Ai = {<x, µa(x), νa(x)>/x B} IFI(B). Then Ai and <U i Ai> are IFI(B). Hence IFI (B) is a sublattice of the lattice of IFSR(B). Consequently <IFI (B),, > forms a complete lattice where A B = <A B> and A B = A B. Theorem 6.2.The set of intuitionistic fuzzy subrings IFSR (B) possessing sup property is a sublattice of IFSR (B). Proof. Clearly follows. Theorem 6.3.The set of int. fuzzy ideals IFI (B) each member of which possess sup property is a sublattice of the lattice of IFSR (B). Proof. Clearly holds, as intersection of two sublattices is a sublattice. Theorem 6.4.The set of characteristic int. fuzzy subrings CIFSR (B) each member of which possesses sup-property is a sublattice of IFI (B) possessing sup property. Proof.By theorem 4.12 the set of CIFSR (B) possessing sup-property is contained in the set of IFI (B) possessing sup-property. For C, D CIFSR (B) each possessing sup-property C D CIFSR (B) possessing sup-property. Also C D CIFS (B) and possess sup-property. Consequently by theorem 5.3 <C D> CIFSR (B). Hence (CIFSR (B),, ) is a lattice where C D = <C D> and C D = C D. Hence CIFSR (B) with sup-property is a sub-lattice of IFI(B) having sup-property.

22 250 K. Meena Figure 1: Lattice structures of sub-lattices of IFSR(R) Let IFSRt(B) denote the set of int. fuzzy sub-rings of B each member of which has the same tip. Theorem 6.5.The set IFSRt (B) is a sub-lattice of IFSR (B). Proof. Clearly IFSRt(B) IFSR(B). Also for P, Q IFSRt(B) P Q = P Q IFSRt(B) and Pi IFSRt(B), i I Pi IFSRt(B). Hence IFSRt (B) is a sub-lattice of IFSR (B). The following results are immediate. Theorem 6.6.The set IFIt (B) of int. fuzzy ideals of B with the same tip t, is sublattice of IFSR (B). Theorem 6.7.The set IFSRst(B) of int. fuzzy subrings of B with same tip t and each member of which possesses sup-property is a sublattice of IFSR(B). Theorem 6.8.The set IFIst(B) of int. fuzzy ideals of B with the same tip t and each member of which possesses sup-property is a sublattice of IFSR(B). Theorem 6.9.The set CIFSR (B) of characteristic int. fuzzy subrings of B with the same tip and each member of which possess sup-property is a sublattice of IFSR (B).

23 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 251 Figure 2: Inter-relationship of sublattices of IFSR (B) Theorem 6.10.The set CIFSR (B) of characteristic intuitionistic fuzzy subrings of B forms a lattice under the ordering of int. fuzzy set inclusion. Proof. The set CIFSR (B) is closed under arbitrary intersection. Also CIFSR (B) contains the greatest element B. Therefore CIFSR (B) is a lattice under the ordering of intuitionistic fuzzy set inclusion. Corollary 6.11.The set of CIFSR (B) of characteristic int. fuzzy subrings of B is a sublattice of the lattice of int. fuzzy ideals of B. 7 CONCLUSION The main objective of study in this paper is the characteristic intuitionistic fuzzy subring of an intuitionistic fuzzy subring. It is proved that the inf-supstar family of characteristic intuitionistic fuzzy sets is a lattice. More precisely it follows that it is a sublattice of the lattice of intuitionistic fuzzy ideals. Moreover various sublattices of intuitionistic fuzzy subrings are constructed. REFERENCES [1] Adam Niewiadomski and Eulalia Szmidt: Handling Uncertainly in natural sentence via IFS, Notes on Intuitionistic Fuzzy Sets, 7 (2001), No,

24 252 K. Meena [2] AfsharAlam, Sadia Hussian and Yasir Ahmad: Applying Intuitionistic Fuzzy Approach to Reduce Search Domain in a Accidental Case, IJACSA, 1, No. 4 (2010). [3] N. Ajmal and K. V. Thomas: Fuzzy Lattices, Information Science, 79 (1994) [4] N. Ajmal and K. V. Thomas: The lattices of fuzzy ideals of a ring, Fuzzy Sets and Systems 74 (1995), [5] N. Ajmal and I. Jahan: A Study of Normal Fuzzy Subgroups and Characteristic Fuzzy Subgroups of a Fuzzy Group, Fuzzy Inf. Engineering 2 (2012) [6] N. Ajmal: (1996) Fuzzy groups with sup-property. Inform. Sci. 93: [7] N. Ajmal: (1995), The Lattice of Fuzzy Normal Subgroups is Modular, Inform. Sci. 83, [8] N. Ajmal, A. Jain (2009) Some constructions of the joint of fuzzy subgroups and certain lattices of fuzzy subgroups with sup-property, Inform. Sci. 179, [9] K. T. Atanassov: Intuitionistic Fuzzy Sets. VII ITKRs Session, Sofia, June 1983 (Deposed in Central Sci.-Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian) [10] D. Boyadzhieva: B. Kolev and N. Netov, Intuitionistic fuzzy data warehouse and some analystical operations, IS, 5th IEEE, International Conference (2010), [11] B. Banerjee and D. K. Basnet: Intuitionistic Fuzzy Subrings and ideals, J. Fuzzy Math., Vol. 11, No. 1, 2003, [12] R. Biswas: Intuitionistic fuzzy subgroups, Mathematical Forum, Vol. X, (1989), [13] V. N. Dixit, R. Kumar and N. Ajmal: Level subgroups and union of fuzzy subgroups. Fuzzy Sets and Systems 37: (1991) [14] Eulalia Szmidt and JanuzKaprzyk: Inter - Fuzzy set in some Medical Applications, NIFS, 7 (2001), 4, [15] K. C. Gupta and B. K. Sharma: Operator domains on fuzzy groups. Inform. Sci. 84: (1995) [16] D. S. Kim: Characteristic fuzzy groups. Commun. Korean Math. Soc. 18: (2003)

25 Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 253 [17] Magdalena Rencova: An example of Applications of IFS to sociometry, Bulgarian Academy of Sciences, Cybernetics and Technologies, Vol 9, No.2 (2009). [18] L. Martinez: (1995) Fuzzy subgroups of fuzzy groups and fuzzy ideals of fuzzy rings. J. Fuzzy Math. 3, [19] L. Martinez: (1999) Prime and Primary L-fuzzy ideals and L-fuzzy rings, Fuzzy Sets and Systems, 101, [20] K. Meena and K. V. Thomas: Intuitionistic L-fuzzy Subrings, IMF, 6 (2011), No.52, [21] K. Meena and K. V. Thomas: Intuitionistic L-Fuzzy Rings, GJSFR(F), Vol.12, 2012, No. 14-F, Art.2. [22] K. Meena and K. V. Thomas: Intuitionistic L-Fuzzy Congruence on a Ring, Notes on Intuitionistic Fuzzy Sets, Vol. 19, 2013, No.1, [23] K. Meena and K. V. Thomas: Interval-Valued Intuitionistic Fuzzy Ideals of a Ring, Advances in Fuzzy Sets and Systems, 17(1), 2014, [24] A. Rosenfeld: Fuzzy Groups, J.Math, Anal. Appl., 35 (1971), [25] F. I. Sidky and M. A. A. Mishref: Fully invariant, characteristic and S-fuzzy subgroups. Information Sciences 55: (1991) [26] W. Wu: (1988) Fuzzy Congruences and normal Fuzzy Subgroups, Math. Appl. 1, [27] L. A. Zadeh: Fuzzy sets, Information and Control, 8 (1965), [28] L. A. Zadeh: The concept of a linguistic variable and its application to approximate reasoning. I., Information Sciences, (1975). [29] A. Zaid: On generalized characteristic fuzzy subgroup of finite group, Fuzzy Sets and Systems 43: (1991)

26 254 K. Meena