Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring

Advances in Fuzzy Mathematics. ISSN 0973-533X Volume 12, Number 2 (2017), pp. 229-253 © Research India Publications http://www.ripublication.com

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 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} 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 푦0∈ Y such that 232 K. Meena

푠푢푝푦є푌 µA(y) = µA(푦0) and

푖푛푓푦є푌 νA(y) = νA(푦0). 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

. −1 −1 −1 The int. inverse image of Y , 푓 (B) = {x, 푓 (µB)(x), 푓 (νB)(x)˃/x ∈ X} is defined −1 −1 as 푓 (µB)(x) = µB(f(x)), 푓 (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 2.10. [7] Let {Ai} ∈ IFS(R). Then 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) 퐴푡 is a subring of R, ∀t ∈ [0, 1].

Proposition 2.12.[7] Let A ∈ IFS(R) with tip t0 = µA(0). Then

> > ˂퐴푡˃ = ˂퐴푡 ˃ ∀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) 퐴푡 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). 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:

˂휇퐴˃퐵= ∩ {µ ∈ IFSR (B): µA ⊆ µ}

˂훾퐴˃퐵= ∩ {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 퐵푡 ,∀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, µA(xy ) ≥ min(µA(x),µB(y )) ≥ t and

∗ ∗ 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 = ∪푟>푡 퐴푟and 퐵푡 = ∪푟>푡Br. Clearly > > A t is an ideal of 퐵푡 . 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).

1 1 Let t = µA(x + y) implies x + y ∈ 휇퐴푡1 and t ∈ Im µA. Also

1 1 1 t < µA(x) ∧ µA(y) ⇒ µA(x) >푡 , µA(y) > t > ⇒ x, y 휇퐴푡1 > But 푥 + 푦휇퐴푡1 is a contradiction.   > > ∗ If 푡 = 휇 (푥푦) 푡 <µ (x) ∧ µA(y)  x 휇 ,y 휇 and t ∈ ImµA. 퐴 퐵 퐵 푡 퐴 푡 But 푥푦휇> is a contradiction. 퐴푡∗ 1 1 Similarly, if 푡 = νA(x + y) implies x + y ∈ 훾퐴푡1 and 푡 ∈ ImνA. Also

1 푡 >νA(x) ∨ νA(y) 1 1 ⇒ νA(x) < 푡 , νA(y) < 푡 ⇒ x, y∈ 휇> 퐴푡1 But 푥 + 푦휈> . This is a contradiction. 퐴푡1

If t2 = νA (xy) then t2 >νA(x) ∨ νB(y)

⇒ νA (x) < t2, νB(y) < t2. > > ⇒푥 ∈ 휈 , 푦 ∈ 휈 and t2 ∈ ImνA. 퐴푡2 퐵푡2 But 푥푦휈> is a contradiction. 퐴푡2 > > 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), 1 t = µB(r).

236 K. Meena

Case i: 1 1 1 1 Let t ≥ t . This implies that t0 ≥ µA(x) ≥ t = µB(r). Thus t ∈ Im µB and t ≤ t0. Since 1 At is an ideal of 퐵푡 , xr ∈휇퐴푡1, i.e.

1 µA (xr) ≥ t ≥ µA(x) ∧ µB(r)

1 1 Also let t = µA(x), t = µA(y), x,y∈ R. Then, t0 ≥ t = µA(x) ≥ µA(y) = t and t ≤ t0.

Therefore x ∈ µAt, x ∈ µAt1 and y ∈ µAt1

1 µA(x + y) ≥ t = 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: 1 Let t ≤ t . 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). 1 Therefore µA(xr) ≥ µA(x)∧µB(r). Also let t = µA(x), t = µA(y), x,y∈ R. Then

1 t0 ≥ µA(x) = t ≤ µA(y) = t

1 ⇒ t = µA(x) ≤ µA(y) = t .

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 퐵푡 . Hence by theorem 3.7 ˂A˃ ∈ IFI (B).

Proposition 3.9.Let {Ai} ⊆ IFI (B) be any family. Then

(i) ∪푖Ai∈ IFI (B). Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 237

(ii) ∩푖Ai ∈ 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 ∪푖(Ai) t = (∪푖Ai) t . The > > clearly (∪푖Ai) t is an ideal of Bt . Hence by theorem 3.7, ∪푖Ai ∈ IFI (B).

(ii) Let t0 = sup {∩푖Ai}. Then t0 = inf{sup Ai}. Also ∩푖Ai ∈ IFS (B). Let i > t∈ [0,t0]. Since Ai ∈ IFI(B), and by theorem3.7 for each i, (퐴푖 )푡 is an ideal of > > > Bt . By proposition 2.10 ∩푖 (퐴푖)푡 = (∩푖 퐴푖)푡

> > Then clearly (∩푖 퐴푖)푡 is an ideal of 퐵푡 . Hence by theorem 3.7 ∩푖Ai ∈ 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].

238 K. Meena

Proof. Necessary Part: Let t ∈ [0,t0] and Q ∈ CIFS(B). Then for x ∈ (µB)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.

Similarly for x ∈ (νB)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. 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 휇퐵= b0 then b0 ∈ B and if 푖푛푓훾퐵= b1 then b1 ∈ B. Definition 4.5.A non-empty subset Y of the unit interval [0, 1] is said to be a inf- supstar 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 2. 1. 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

∪푖ImAi is a inf-supstar subset.

Proposition 4.7.Let {퐴푖}푖є퐼∈ IFS (B) be an inf-supstar family. Then

i ) Ai ∈ IFSs(B) for each i∈ I.

(ii) ∪푖∈훺 퐴푖Є퐼퐹푆푠(퐵)whereΩ ⊆ I.

Proof. (i) Given {퐴푖}푖є퐼is a inf-supstar family. Then ∪푖 퐼푚푔퐴푖 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 ∪푖ImAi is a inf-superstar subset. Hence every subset of ∪푖 퐼푚푔퐴푖 is closed under arbitrary infimum and supremum. Therefore Ai ∈ IFSs (B) for all i∈ I.

(i) Given {퐴푖}푖є퐼is a inf-supstar family. Then∪푖 퐼푚푔퐴푖 is a inf-supstar subset ⇒ ∪푖ImµAi is closed under arbitrary unions. Let Ω ⊂ I. Then for x ∈ R

Im∪푖∈Ω µAi = ∪푥∈푅 {(∪푖∈ΩµAi) (x)}

= ∪푥∈푅{푠푢푝푖є훺(µAi(x))}

= ∪푥∈푅 휇퐴푥where 휇퐴푥= {µAi(x)/i∈Ω}.

∈ Then µ퐴푥= {µAi(x)/i Ω} ⊆ ∪ ImµAi. But every subset of ∪푖є퐼 퐼푚푔휇퐴푖 is closed under arbitrary unions. Hence

sup µ퐴푖∈ µ퐴푥⊂ ∪푖є퐼 퐼푚푔휇퐴푖 Thus

Im∪푖∈퐼 µ퐴푖⊂∪푖∈퐼 퐼푚µ퐴푖. (1) 240 K. Meena

Similarly given {Ai}i∈I is a inf-supstar family. Then∪푖є퐼 퐼푚퐴푖 is a inf-supstar subset ⇒ ∩푖∈퐼ImνAi is closed under arbitrary intersection.

Let Ω ⊂ I. Then for x ∈ R,

Im ∩푖∈훺 휈퐴푖 = ∪푥∈푅{(∩푖∈훺 휈퐴푖(x)}

= ∪푥∈푅{inf(휈퐴푖(x)/i∈Ω)}

= ∪푥∈푅 {푖푛푓휈퐴푥} where 훾퐴푥= {νAi(x)/i∈Ω}.

∈ Then 훾퐴푥 = {νAi(x) /i Ω} ⊆ ∪푖∈퐼ImνAi. Since every subset of ∪Im 훾퐴푖is closed under arbitrary intersection

inf 휈퐴푥∈ ⊆ 휈퐴푥 ∪푖∈퐼Im휈퐴푖 Thus

퐼푚푖∈훺∩휈퐴푖 ⊆ ∪푖∈퐼Im휈퐴푖 (2)

Hence from (1) and (2) ∪푖є훺Ai ∈ 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 ∈ 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{퐴푖}, I= {1,..., n} be a inf-sup star family. Then ∪푖∈퐼Im Ai is a inf-supstar subset which implies that ∪푖∈퐼Ai 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∪푖∈퐼Ai is a inf-supstar subset

⇒∪푖∈퐼Ai is a inf-supstar subset

⇒{퐴푖}푖є퐼 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{퐴푖}푖є퐼is a inf-sup-star family implies ∪ ImAi is a inf-supstar subset. Also 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 ˅푖є퐼µ퐴푖 = 푡표 and ∧푖є퐼 ˅퐴푖 = 푡1then 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 ∈ (휇퐴1)t ∨ (휇퐴2)t ∨ ... and

x ∈ (훾퐴1)t ∧ (훾퐴2)t ∧ ... such that (휇퐴1 (x)∨ 휇퐴2 (x)∨...) ≥ t and (훾퐴1(x)∧훾퐴2(x)∧...) ≤ t. Therefore

x∈ (˅ µ퐴푖) t and x∈ (∧˅퐴푖) t. Hence x ∈ (∪Ai) t. Therefore

x ∈ ∪(Ai)t ⊆ (∪Ai)t. (4)

Then from (3) and (4) (∪Ai) t = ∪ (Ai) t.

Theorem 4.10.If {Ai} ∈ IFS (B) is a maximal inf-supstar family then {퐴푖}푖є퐼 is a complete lattice under the order of fuzzy set inclusion.

Proof. Let{퐴푖}푖є퐼∈ IFS (B) be a maximal inf-supstar family. Then ∪푖∈퐼Im Ai is a inf-supstar subset ⇒ ∪푖∈퐼Imµ퐴푖 , ∪푖∈퐼Im휈퐴푖are inf-supstar subsets. Now to show that{퐴푖}푖є퐼 is closed under arbitrary supremum and infimum.

Let Ω ⊂ I. To show that ∪푖∈훺 µ퐴푖 ∈ {퐴푖} and ∩푖∈훺 휈퐴푖 ∈ {퐴푖} . Since {Ai} is a inf- supstar family 퐼푚푖∈훺 ∪ µ퐴푖 ⊆ ∪푖∈퐼Im µ퐴푖 ,Hence

퐼푚푖∈훺 ∪ µ퐴푖 ∪ [∪푖∈퐼Im µ퐴푖 ]=∪푖∈퐼Im µ퐴푖 , (5)

Also ∪푖∈퐼Im µ퐴푖 ,is a inf-supstar subset. Hence 퐼푚푖∈훺 ∪ µ퐴푖 ∪ [∪푖∈퐼Im µ퐴푖 ] is a inf-supstar subset. Thus ∪푖∈훺 µ퐴푖 푈 {µ퐴푖 : 푖 ∈ I} is a inf-supstar family. By the maximality of inf-supstar family,

∪푖∈훺 µ퐴푖 ∪ {µ퐴푖 : 푖 ∈ I} = {µ퐴푖 : 푖 ∈ I} .

Therefore, ∪푖∈훺 µ퐴푖 ∈ {µ퐴푖 : 푖 ∈ I}. Similarly, since {퐴푖}푖є퐼 is a inf-supstar family it implies that

퐼푚 ∩푖∈퐼 휐퐴푖 ∪푖∈퐼Im 휐퐴푖 ,. Hence 242 K. Meena

(퐼푚 ∩푖∈퐼 휐퐴푖) ∪ [∪푖∈퐼Im 휐퐴푖 ,] = ∪푖∈퐼Im 휐퐴푖 , (6)

Clearly ∪ Im νAI is a inf-supstar subset. Hence (퐼푚 ∩푖∈퐼 휐퐴푖) ∪ [∪푖∈퐼Im 휐퐴푖 ,] is inf-supstar subset. Thus ∩푖∈훺 휐퐴푖 푈 {휐퐴푖 : 푖 ∈ I} is a inf-supstar family. By the maximality of inf-supstar family

∩푖∈훺 휐퐴푖 ∪ {휐퐴푖 : 푖 ∈ I} = {휐퐴푖 : 푖 ∈ I} (7)

Therefore ∩푖∈훺 휐퐴푖 ∈ {휐퐴푖 : 푖 ∈ I}. It follows that {Ai}i∈I is a complete lattice under the ordering of fuzzy set inclusion.

Theorem 4.11. Let {퐴푖}푖є퐼 ⊂ CIFS (B). Then

(i) ∩Ai ∈ CIFS (B).

(ii) ∪Ai ∈ CIFS (B) provided {Ai} is a inf-supstar family.

Proof.(i) Given {퐴푖}푖є퐼 ⊂ CIFS(B) ⇒ µ퐴푖 (Tx) ≥ µ퐴푖 (x), T ∈ A(µB)t ∀t ∈ [0,t0] and

휐퐴푖(Tx) ≤ 휐퐴푖 (x), T ∈ A(νB)t ,∀ t ∈ [0,t0] where t0 is the tip of Ai. Clearly, ( ) ∩Ai = {/ x ∈ R} and ∩푖∈퐼 퐴푖 ∈ IFS B .

Let t0 = sup (∩푖 µ퐴푖 ) and t ∈ [0,t0].

Then t0 = inf sup (µ퐴푖 ) so that t ≤ t0 ≤ sup µAi ∀ i∈ I. Let x ∈ (µB) t , T ∈ A(µB)t , t∈ [0,t0]

∩푖∈퐼µAi(Tx)=inf µ퐴푖 (Tx) ≥ inf µAi(x) (8) =∩푖∈퐼 µAi(x)

Similarly let ∪푖∈퐼 휐퐴푖∈ IFS (B). Let t0 = inf(∪푖 휐퐴푖) and t ∈ [0.t0]. Then

t0 = sup inf(휐퐴푖) so that t ≤ inf 휐퐴푖≤ t0 for each i∈ I. Now for x ∈ (νB) t,

T∈ A (νB)t, ( ) ∪푖∈퐼 휐퐴푖 푇푥 = 푠푢푝 휐퐴푖(푇푥)

≤ sup 휐퐴푖(x) (9)

= ∪푖∈퐼 휐퐴푖 (x).

Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 243

From (8) and (9)

⇒ ∩푖∈퐼 Ai ∈ CIFS (B)

(ii) For i∈ I, ∪ Ai = {/x ∈ R}. Clearly ∪푖∈퐼 µ퐴푖 ∈ IFS (B)

and let t0 = sup{ ∪ µ퐴푖 }. Then t0 = sup [sup µ퐴푖 ]. Let t ∈ [0,t0]. Given

{Ai} ∈ CIFS (B) ⇔ {µ퐴푖 }t,, {휐퐴푖}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., (∪ µ퐴푖 ) t = ∪(µ퐴푖 ) 푡and (∩ 휐퐴푖)푡= ∩(휐퐴푖)푡

Since arbitrary union and intersection of characteristic subsets of a ring R is a characteristic subset, (∪ µ퐴푖 )t and (∩ 휐퐴푖)푡 are characteristics subsets of Bt. By theorem 4.2. ∪푖∈퐼 퐴푖∈ CIFS (B).

Theorem 4.12. Let A ∈ CIFS (B). Then A ∈ IFI (B).

Proof. Let x, r ∈ R. Take 푡표 = µB(x) ∧ µA(r). ⇒ 푡표≤ supµ퐴, x∈(µ퐵)푡표, r ∈(µ퐴)푡표. Define

푇푟: (휇퐵) 푡표→ (휇퐵) 푡표 є Tr(x) = xr.

Then 푇푟 ∈ A [(µB)t0].

Since A ∈ CIFS (B) ⇒ (휇퐴) 푡표is a characteristic subset of (휇퐵) 푡표

⇒ Tr(휇퐴) 푡표⊆(휇퐴) 푡표 ∀ Tr∈ A(휇퐵) 푡표

Since x ∈(휇퐵) 푡표 , Tr(x) ∈ (휇퐴) 푡표 ⇒ 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: (휇퐵) 푡1 → (휇퐵) 푡1є Ty(xy) = x + y.

Then Ty ∈(휇퐵) 푡1. Since A ∈ CIFS (B) ⇒ (휇퐴) 푡1is a characteristic subset of

(휇퐵) 푡1⇒ Ty(휇퐴) 푡1⊆(휇퐴) 푡1 , ∀ Ty ∈ A(휇퐵) 푡1. Since

x∈(휇퐴) 푡1, Ty x∈(휇퐴) 푡1.

⇒µA(푇푦x) ≥ t1 = µA(x) ∧ µA(y)

⇒µA(x + y) ≥ µA(x) ∧ µA(y). 244 K. Meena

For x, r ∈ R let t2 = νB(x) ∨ νA(r) ⇒ x ∈(휈퐵)푡2r ∈(휈퐴)푡2 ,t2 ≥ inf νA.

Define Tr: (훾퐵) 푡2 →(훾퐵) 푡2 є Tr(x) = xr. Then Tr ∈ A(휈퐵)푡2. Since A ∈ CIFS (B),

(휈퐴)푡2is a characteristic subset of (휈퐵)푡2 ⇒ Tr (휈퐴)푡2⊆(휈퐴)푡2 ∀Tr∈ A(휈퐵)푡2

Since x ∈(휈퐵)푡2, Tr(x) ∈ (휈퐴)푡2

⇒ 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 ∈ (휈퐴)푡3 y ∈(훾퐵) 푡3 and t3 ≤ inf νA.

Define Ty: (훾퐵) 푡3→(훾퐵) 푡3 є Ty(x) = x + y.

Clearly Ty ∈ A(훾퐵) 푡3. Since A ∈ CIFS (B) implies (훾퐴) 푡2is a characteristic subset of (훾퐵) 푡3 . ⇒ Ty (훾퐵) 푡3 ⊆(훾퐵) 푡3, ∀ Ty ∈ A(훾퐵) 푡2.

Since x ∈(훾퐴) 푡2 , Ty(x) ∈ (훾퐴) 푡3

⇒ νA(Ty(x)) ≤ t3 = νA(x) ∨ νA(y)

⇒ νA(x + y) ≤ νA(x) ∨ νA(y). Hence A ∈ IFI (B).

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) = 푠푢푝푇푥=푦 [µA/(µB)t(x)]

= sup [µA(x)] Tx=y Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 245

−1 = µA(T (y)) (∵ T ∈ A(µB)t)

≥ µA(y) (∵ A ∈ CIFS(B))

= µA/(µB)푡 (10)

Similarly, for y ∈ (µB)푡

[µA/ (µB)t](y) = µA(y) −1 = µA(TT (y)) [∵ T ∈ A(µB)t] −1 ≥ µA(T (y)) [∵ A ∈ CIFS(B)] −1 −1 = (푇 ) 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) = 푖푛푓푇푥=푦(νA/(νB)t(x)) =푖푛푓푇푥=푦(휈퐴(푥)

−1 = νA(T (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) −1 = νA [TT (y)]

−1 ≤ νA [T (y)] (∵ A ∈ CIFS(B)) −1 −1 =훾퐴 (푇 ) (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 ∈ 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. 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.

n Example. Consider the ring R = (푍4 , +, ·) where 푍4 = {0, 1, 2, 3} and let (2 ) be the integral multiples of 2 where n is a fixed natural number.

Let A = {/x ∈푍4 } be an IFS(푍4 ) defined as

Let B = {/x ∈푍4 } be an IFS (푍4 ) defined as

Then A and B are IFSR (푍4 ) 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(A표B)(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 표B)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 표B)t is a characteristic subset of Pt ⇒ A표 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). 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 ∈ B} and D = {: 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

ˆ A = {/x ∈ R}, where µˆA(x) = sup{r : x ∈ <(µA)r>} and νˆA(x) = inf{r : x ∈ <(νA)r>}. Then Aˆ ∈ IFSR (B) and Aˆ = . Theorem5.2. [8] Let A ∈ IFSR (B) and possesses sup-property. Then possesses sup-property. Theorem 5.3.Let A ∈ CIFS (B) and possesses sup-property. Then ∈ CIFSR (B) having sup-property. Proof. Suppose is not an CIFSR (B) ⇒ ∉ CIFSR (B) ⇒ = {, <νA>>} ∉ CIFSR (B). Then there exists 푡 ∈ [0, 휇 (0)] ϶ < 휇 > is not a characteristic subring of 표 퐵 퐴푡0

(휇퐵)푡표⇒ 푇표∈ A(휇퐵)푡표 є 푇표 (< 휇퐴 >푡표) ⊄ < 휇퐴 >푡표

Hence there exists a y0 ∈ R y0 ∉ <µA>t0 but y0 ∈ T0(< 휇퐴 >)푡표. Thus 248 K. Meena

y0 =T0(x0) where x0 ∈ < 휇퐴 >푡표

⇒<µA>(x0) ≥ 푡표

⇒ sup{r : x0 ∈ <(µA)r>} ≥ t0. r∈Imµa

Since A possesses sup-property,

⇒ ∃ r0 ∈ Im 휇퐴 є 푥표 ∈ <(휇퐴) 푟표> and r0 ≥ t0 (14)

As 푥표 ∈< (휇퐴)푟표 > we get x0 = a0a1 ...an where ai ∈ (휇퐴)푟표

Hence T0푥표 = 푇0(a0a1 ...an) = 푇0a0 · 푇0a1 ... 푇0an.

Also as µA ∈ CIFS (B), µA (푇0ai) ≥ µA (ai) and since ai ∈ (휇퐴)푟0

⇒ 푇0푎푖 ∈(휇퐴)푟0.