Characteristic Intuitionistic Fuzzy Subrings of an …Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring K. Meena BS & H Department, Muthoot Institute of

Advances in Fuzzy Mathematics.

ISSN 0973-533X Volume 12, Number 2 (2017), pp. 229-253

© Research India Publications

http://www.ripublication.com

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

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 Aii∈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 𝑦0∈ Y such that

232 K. Meena

𝑠𝑢𝑝𝑦є𝑌 µA(y) = µA(𝑦0) and

𝑖𝑛𝑓𝑦є𝑌 νA(y) = νA(𝑦0).

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 2.10. [7] Let Ai ∈ IFS(R). Then


(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) 𝐴𝑡> is a subring of R, ∀t ∈ [0, 1].

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

˂𝐴𝑡˃> = ˂𝐴𝑡>˃ ∀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) 𝐴𝑡> 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 ∈ Rbe 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:

˂𝜇𝐴˃𝐵= ∩ µ ∈ IFSR (B): µA ⊆ µ

˂𝛾𝐴˃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 𝐵𝑡

>,∀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 𝐴𝑟and 𝐵𝑡

>= ∪𝑟>𝑡Br. Clearly

A>t is an ideal of 𝐵𝑡

>.


iii⇒ iv Obvious.

iv⇒ i

Let A>t be an ideal of Bt

>, ∀t ∈ ImA. Suppose AIFI (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 t1 = µA(x + y) implies x + y ∈ 𝜇𝐴𝑡1 and t1 ∈ Im µA. Also

t1 < µA(x) ∧ µA(y) ⇒ µA(x) >𝑡1, µA(y) > t1

⇒ x, y 𝜇𝐴𝑡1>

But 𝑥 + 𝑦𝜇𝐴𝑡1

> is a contradiction.

If 𝑡 = 𝜇𝐴 (𝑥𝑦) 𝑡 <µ𝐵(x) ∧ µA(y) x 𝜇𝐵 𝑡> ,y 𝜇𝐴 𝑡

> and t∗ ∈ ImµA.

But 𝑥𝑦𝜇𝐴𝑡∗> is a contradiction.

Similarly, if 𝑡1 = νA(x + y) implies x + y ∈ 𝛾𝐴𝑡1 and 𝑡1

∈ ImνA. Also

𝑡1 >νA(x) ∨ νA(y)

⇒ νA(x) < 𝑡1 , νA(y) < 𝑡1

⇒ x, y∈ 𝜇𝐴𝑡1

>

But 𝑥 + 𝑦𝜈𝐴𝑡1

> . This is a contradiction.

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

⇒ νA (x) < t2, νB(y) < t2. ⇒𝑥 ∈ 𝜈𝐴𝑡2

> , 𝑦 ∈ 𝜈𝐵𝑡2

> and t2 ∈ ImνA.

But 𝑥𝑦𝜈𝐴𝑡2

> is a contradiction.

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), t1 = µB(r).

236 K. Meena

Case i:

Let t ≥ t1. This implies that t0 ≥ µA(x) ≥ t1 = µB(r). Thus t1 ∈ Im µB and t1 ≤ t0. Since

At is an ideal of 𝐵𝑡1, xr ∈𝜇𝐴𝑡1, i.e.

µA (xr) ≥ t1 ≥ µA(x) ∧ µB(r)

Also let t = µA(x), t1 = µA(y), x,y∈ R. Then, t0 ≥ t = µA(x) ≥ µA(y) = t1 and t ≤ t0.

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

µA(x + y) ≥ t1 = 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 ≤ t1. 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), t1 = µA(y), x,y∈ R. Then

t0 ≥ µA(x) = t ≤ µA(y) = t1

⇒ t = µA(x) ≤ µA(y) = t1.

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 𝐵𝑡

>.

Hence by theorem 3.7 ˂A˃ ∈ IFI (B).

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

(i) ∪𝑖Ai∈ IFI (B).


(ii) ∩𝑖Ai ∈ 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 = (∪𝑖Ai)

>t . The

clearly (∪𝑖Ai)>

t is an ideal of Bt>. Hence by theorem 3.7, ∪𝑖Ai ∈ IFI (B).

(ii) Let t0 = sup ∩𝑖Ai. Then t0 = infsup Ai. Also ∩𝑖Ai ∈ IFS (B). Let i t∈ [0,t0]. Since Ai ∈ IFI(B), and by theorem3.7 for each i, (𝐴𝑖 )𝑡

>is an ideal of

Bt>. By proposition 2.10 ∩𝑖 (𝐴𝑖)𝑡

>= (∩𝑖 𝐴𝑖)𝑡>

Then clearly (∩𝑖 𝐴𝑖)𝑡>is an ideal of 𝐵𝑡

>. Hence by theorem 3.7 ∩𝑖Ai ∈ 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.


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 𝜇𝐵= b0 then b0 ∈ B and if 𝑖𝑛𝑓𝛾𝐵= 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 Aii∈I∈ IFS(R). Then Aii∈Iis said to be inf-supstar family if ∪𝑖ImAi is a inf-supstar subset.

Proposition 4.7.Let 𝐴𝑖𝑖є𝐼∈ IFS (B) be an inf-supstar family. Then

i ) Ai ∈ IFSs(B) for each i∈ I.

(ii) ∪𝑖∈𝛺 𝐴𝑖Є𝐼𝐹𝑆𝑠(𝐵)whereΩ ⊆ I. Proof. (i) Given 𝐴𝑖𝑖є𝐼is a inf-supstar family. Then ∪𝑖 𝐼𝑚𝑔𝐴𝑖 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 ∪𝑖ImAi is a inf-superstar subset. Hence every subset

of ∪𝑖 𝐼𝑚𝑔𝐴𝑖 is closed under arbitrary infimum and supremum. Therefore

Ai ∈ IFSs (B) for all i∈ I. (i) Given 𝐴𝑖𝑖є𝐼is a inf-supstar family. Then∪𝑖 𝐼𝑚𝑔𝐴𝑖 is a inf-supstar subset ⇒

∪𝑖ImµAi is closed under arbitrary unions. Let Ω ⊂ I. Then for x ∈ R

Im∪𝑖∈Ω µAi = ∪𝑥∈𝑅 (∪𝑖∈ΩµAi) (x)

= ∪𝑥∈𝑅𝑠𝑢𝑝𝑖є𝛺(µAi(x))

= ∪𝑥∈𝑅 𝜇𝐴𝑥where 𝜇𝐴𝑥

= µAi(x)/i∈Ω.

Then µ𝐴𝑥= µAi(x)/i∈Ω ⊆ ∪ ImµAi. But every subset of ∪𝑖є𝐼 𝐼𝑚𝑔𝜇𝐴𝑖

is

closed under arbitrary unions. Hence

sup µ𝐴𝑖∈ µ𝐴𝑥

⊂ ∪𝑖є𝐼 𝐼𝑚𝑔𝜇𝐴𝑖

Thus

Im∪𝑖∈𝐼 µ𝐴𝑖⊂∪𝑖∈𝐼 𝐼𝑚µ𝐴𝑖

. (1)

240 K. Meena

Similarly given Aii∈I is a inf-supstar family. Then∪𝑖є𝐼 𝐼𝑚𝐴𝑖 is a inf-supstar subset

⇒ ∩𝑖∈𝐼ImνAi is closed under arbitrary intersection.

Let Ω ⊂ I. Then for x ∈ R,

Im ∩𝑖∈𝛺 𝜈𝐴𝑖 = ∪𝑥∈𝑅(∩𝑖∈𝛺 𝜈𝐴𝑖

(x)

= ∪𝑥∈𝑅inf(𝜈𝐴𝑖(x)/i∈Ω)

= ∪𝑥∈𝑅 𝑖𝑛𝑓𝜈𝐴𝑥 where 𝛾𝐴𝑥

= νAi(x)/i∈Ω.

Then 𝛾𝐴𝑥 = νAi(x) /i∈Ω ⊆ ∪𝑖∈𝐼ImνAi. Since every subset of ∪Im 𝛾𝐴𝑖is closed

under arbitrary intersection

inf 𝜈𝐴𝑥∈ ⊆ 𝜈𝐴𝑥

∪𝑖∈𝐼Im𝜈𝐴𝑖

Thus

𝐼𝑚𝑖∈𝛺∩𝜈𝐴𝑖 ⊆ ∪𝑖∈𝐼Im𝜈𝐴𝑖

(2)

Hence from (1) and (2) ∪𝑖є𝛺Ai ∈ 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. Let𝐴𝑖, I= 1,..., n be a inf-sup star family. Then ∪𝑖∈𝐼Im Ai is

a inf-supstar subset which implies that ∪𝑖∈𝐼Ai 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∈𝐼Ai is a inf-supstar subset ⇒∪𝑖∈𝐼Ai is a inf-supstar subset

⇒𝐴𝑖𝑖є𝐼 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. Given𝐴𝑖𝑖є𝐼is a inf-sup-star family implies ∪ ImAi is a inf-supstar

subset. Also


(∪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 ˅𝑖є𝐼µ𝐴𝑖

= 𝑡𝑜 and ∧𝑖є𝐼 ˅𝐴𝑖= 𝑡1then t0 ,t1∈ ImAi where t0 ≥ t and t1 ≤ t.

Hence

x∈ ∪(Ai)t ⇒ (∪Ai)t ⊆ ∪(Ai)t

Conversely let x ∈ ∪ (Ai)t. Then

x ∈ (𝜇𝐴1)t ∨ (𝜇𝐴2

)t ∨ ... and

x ∈ (𝛾𝐴1)t ∧ (𝛾𝐴2

)t ∧ ...

(3)

such that (𝜇𝐴1 (x)∨ 𝜇𝐴2

(x)∨...) ≥ t and (𝛾𝐴1(x)∧𝛾𝐴2

(x)∧...) ≤ t. Therefore

x∈ (˅ µ𝐴𝑖) t and x∈ (∧˅𝐴𝑖

) 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 𝐴𝑖𝑖є𝐼 is a complete lattice under the order of fuzzy set inclusion. Proof. Let𝐴𝑖𝑖є𝐼∈ IFS (B) be a maximal inf-supstar family. Then ∪𝑖∈𝐼Im Ai

is a inf-supstar subset ⇒ ∪𝑖∈𝐼Imµ𝐴𝑖 , ∪𝑖∈𝐼Im𝜈𝐴𝑖are inf-supstar subsets.

Now to show that𝐴𝑖𝑖є𝐼 is closed under arbitrary supremum and infimum.

Let Ω ⊂ I. To show that ∪𝑖∈𝛺 µ𝐴𝑖∈ 𝐴𝑖 and ∩𝑖∈𝛺 𝜈𝐴𝑖

∈ 𝐴𝑖 . Since Ai is a inf-

supstar family 𝐼𝑚𝑖∈𝛺 ∪ µ𝐴𝑖 ⊆ ∪𝑖∈𝐼Im µ𝐴𝑖 ,Hence

𝐼𝑚𝑖∈𝛺 ∪ µ𝐴𝑖∪ [∪𝑖∈𝐼Im µ𝐴𝑖 ]=∪𝑖∈𝐼Im µ𝐴𝑖 , (5)

Also ∪𝑖∈𝐼Im µ𝐴𝑖 ,is a inf-supstar subset. Hence 𝐼𝑚𝑖∈𝛺 ∪ µ𝐴𝑖∪ [∪𝑖∈𝐼Im µ𝐴𝑖 ] is a

inf-supstar subset. Thus ∪𝑖∈𝛺 µ𝐴𝑖 𝑈 µ𝐴𝑖 : 𝑖 ∈ I is a inf-supstar family. By the

maximality of inf-supstar family,

∪𝑖∈𝛺 µ𝐴𝑖 ∪ µ𝐴𝑖 : 𝑖 ∈ I = µ𝐴𝑖 : 𝑖 ∈ I .

Therefore, ∪𝑖∈𝛺 µ𝐴𝑖 ∈ µ𝐴𝑖 : 𝑖 ∈ I.

Similarly, since 𝐴𝑖𝑖є𝐼 is a inf-supstar family it implies that

𝐼𝑚 ∩𝑖∈𝐼 𝜐𝐴𝑖 ∪𝑖∈𝐼Im 𝜐𝐴𝑖 ,. Hence

242 K. Meena

(𝐼𝑚 ∩𝑖∈𝐼 𝜐𝐴𝑖) ∪ [∪𝑖∈𝐼Im 𝜐𝐴𝑖 ,] = ∪𝑖∈𝐼Im 𝜐𝐴𝑖 ,

(6)

Clearly ∪ Im νAI is a inf-supstar subset. Hence (𝐼𝑚 ∩𝑖∈𝐼 𝜐𝐴𝑖) ∪ [∪𝑖∈𝐼Im 𝜐𝐴𝑖 ,]

is inf-supstar subset. Thus ∩𝑖∈𝛺 𝜐𝐴𝑖 𝑈 𝜐𝐴𝑖 : 𝑖 ∈ I is a inf-supstar family. By the

maximality of inf-supstar family

∩𝑖∈𝛺 𝜐𝐴𝑖 ∪ 𝜐𝐴𝑖 : 𝑖 ∈ I = 𝜐𝐴𝑖 : 𝑖 ∈ I

(7)

Therefore ∩𝑖∈𝛺 𝜐𝐴𝑖 ∈ 𝜐𝐴𝑖 : 𝑖 ∈ I.

It follows that Aii∈I is a complete lattice under the ordering of fuzzy set inclusion.

Theorem 4.11. Let 𝐴𝑖𝑖є𝐼 ⊂ CIFS (B). Then

(i) ∩Ai ∈ CIFS (B).

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

Proof.(i) Given 𝐴𝑖𝑖є𝐼 ⊂ CIFS(B) ⇒ µ𝐴𝑖 (Tx) ≥ µ𝐴𝑖 (x), T ∈ A(µB)t ∀t ∈ [0,t0] and

𝜐𝐴𝑖(Tx) ≤ 𝜐𝐴𝑖

(x), T ∈ A(νB)t ,∀ t ∈ [0,t0] where t0 is the tip of Ai. Clearly,

∩Ai = <x, ∩ µ𝐴𝑖 ∪𝜐𝐴𝑖>/ x ∈ R and ∩𝑖∈𝐼 𝐴𝑖 ∈ IFS(B).

Let t0 = sup (∩𝑖 µ𝐴𝑖 ) and t ∈ [0,t0].

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

∩𝑖∈𝐼µAi(Tx)=inf µ𝐴𝑖 (Tx)

≥ inf µAi(x)

=∩𝑖∈𝐼 µAi(x)

(8)

Similarly let ∪𝑖∈𝐼 𝜐𝐴𝑖∈ IFS (B). Let t0 = inf(∪𝑖 𝜐𝐴𝑖

) and t ∈ [0.t0]. Then

t0 = sup inf(𝜐𝐴𝑖) so that t ≤ inf 𝜐𝐴𝑖

≤ t0 for each i∈ I. Now for x ∈ (νB) t,

T∈ A (νB)t,

∪𝑖∈𝐼 𝜐𝐴𝑖(𝑇𝑥) = 𝑠𝑢𝑝 𝜐𝐴𝑖

(𝑇𝑥)

≤ sup 𝜐𝐴𝑖(x) (9)

= ∪𝑖∈𝐼 𝜐𝐴𝑖 (x).


From (8) and (9)

⇒ ∩𝑖∈𝐼 Ai ∈ CIFS (B)

(ii) For i∈ I, ∪ Ai = <x, ∪ µ𝐴𝑖 ,∩𝜐𝐴𝑖>/x ∈ R. Clearly ∪𝑖∈𝐼 µ𝐴𝑖 ∈ IFS (B)

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

Ai ∈ CIFS (B) ⇔ µ𝐴𝑖 t,, 𝜐𝐴𝑖t is characteristic subset of (µB)t and(νB)t.

Given Ai ∈ IFS (B) is a inf-supstar family.

(∪Ai) t = ∪ (Ai)t

i.e., (∪ µ𝐴𝑖 ) t = ∪(µ𝐴𝑖 ) 𝑡and (∩ 𝜐𝐴𝑖)𝑡= ∩(𝜐𝐴𝑖

)𝑡

Since arbitrary union and intersection of characteristic subsets of a ring R is a

characteristic subset, (∪ µ𝐴𝑖 )t and (∩ 𝜐𝐴𝑖)𝑡 are characteristics subsets of Bt.

By theorem 4.2. ∪𝑖∈𝐼 𝐴𝑖∈ CIFS (B).

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

Proof. Let x, r ∈ R. Take 𝑡𝑜 = µB(x) ∧ µA(r). ⇒ 𝑡𝑜≤ supµ𝐴, x∈(µ𝐵)𝑡𝑜, r ∈(µ𝐴)𝑡𝑜

. Define

𝑇𝑟: (𝜇𝐵) 𝑡𝑜→ (𝜇𝐵) 𝑡𝑜

є Tr(x) = xr.

Then 𝑇𝑟 ∈ A [(µB)t0].

Since A ∈ CIFS (B) ⇒ (𝜇𝐴) 𝑡𝑜is a characteristic subset of (𝜇𝐵) 𝑡𝑜

⇒ Tr(𝜇𝐴) 𝑡𝑜⊆(𝜇𝐴) 𝑡𝑜

∀ Tr∈ A(𝜇𝐵) 𝑡𝑜

Since x ∈(𝜇𝐵) 𝑡𝑜 , Tr(x) ∈ (𝜇𝐴) 𝑡𝑜 ⇒ 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: (𝜇𝐵) 𝑡1 → (𝜇𝐵) 𝑡1є Ty(xy) = x + y.

Then Ty ∈(𝜇𝐵) 𝑡1. Since A ∈ CIFS (B) ⇒ (𝜇𝐴) 𝑡1

is a characteristic subset of

(𝜇𝐵) 𝑡1⇒ Ty(𝜇𝐴) 𝑡1

⊆(𝜇𝐴) 𝑡1 , ∀ Ty ∈ A(𝜇𝐵) 𝑡1

. Since

x∈(𝜇𝐴) 𝑡1, Ty x∈(𝜇𝐴) 𝑡1

.

⇒µA(𝑇𝑦x) ≥ 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 ∈(𝜈𝐵)𝑡2r ∈(𝜈𝐴)𝑡2

,t2 ≥ inf νA.

Define Tr: (𝛾𝐵) 𝑡2 →(𝛾𝐵) 𝑡2 є Tr(x) = xr. Then Tr ∈ A(𝜈𝐵)𝑡2

. Since A ∈ CIFS (B), (𝜈𝐴)𝑡2

is a characteristic subset of (𝜈𝐵)𝑡2 ⇒ Tr (𝜈𝐴)𝑡2

⊆(𝜈𝐴)𝑡2 ∀Tr∈ A(𝜈𝐵)𝑡2

Since x ∈(𝜈𝐵)𝑡2, Tr(x) ∈ (𝜈𝐴)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 ∈ (𝜈𝐴)𝑡3 y ∈(𝛾𝐵) 𝑡3

and t3 ≤ inf νA.

Define Ty: (𝛾𝐵) 𝑡3→(𝛾𝐵) 𝑡3

є Ty(x) = x + y.

Clearly Ty ∈ A(𝛾𝐵) 𝑡3. Since A ∈ CIFS (B) implies (𝛾𝐴) 𝑡2

is a characteristic subset

of (𝛾𝐵) 𝑡3 . ⇒ Ty (𝛾𝐵) 𝑡3 ⊆(𝛾𝐵) 𝑡3

, ∀ Ty ∈ A(𝛾𝐵) 𝑡2.

Since x ∈(𝛾𝐴) 𝑡2 , Ty(x) ∈ (𝛾𝐴) 𝑡3

⇒ 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) = 𝑠𝑢𝑝𝑇𝑥=𝑦 [µA/(µB)t(x)]

= sup [µA(x)]

Tx=y


= µA(T−1(y)) (∵ T ∈ A(µB)t)

≥ µA(y) (∵ A ∈ CIFS(B))

= µA/(µB)𝑡 (10)

Similarly, for y ∈ (µB)𝑡

[µA/ (µB)t](y) = µA(y)

= µA(TT−1(y)) [∵ T ∈ A(µB)t]

≥ µA(T−1(y)) [∵ A ∈ CIFS(B)]

= (𝑇−1)−1µA(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) = 𝑖𝑛𝑓𝑇𝑥=𝑦(νA/(νB)t(x))

=𝑖𝑛𝑓𝑇𝑥=𝑦(𝜈𝐴(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))

=𝛾𝐴 (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.

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].


Example. Consider the ring R = (𝑍4 , +, ·) where 𝑍4 = 0, 1, 2, 3 and let (2n) be

the integral multiples of 2 where n is a fixed natural number.

Let A = <x, µA(x), νA(x) >/x ∈𝑍4 be an IFS(𝑍4 ) defined as

Let B = <x, µB(x), νB(x)>/x ∈𝑍4 be an IFS (𝑍4 ) defined as

Then A and B are IFSR (𝑍4 ) 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 =

minr, 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).


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, µÂ(x), νÂ(x)>/x ∈ R, where µÂ(x) = supr : x ∈ <(µA)r> and νÂ(x) = infr : 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 𝑡𝑜∈ [0, 𝜇𝐵 (0)] ϶ < 𝜇𝐴𝑡0> is not a characteristic subring of

(𝜇𝐵)𝑡𝑜⇒ 𝑇𝑜∈ A(𝜇𝐵)𝑡𝑜

є 𝑇𝑜 (< 𝜇𝐴 >𝑡𝑜) ⊄ < 𝜇𝐴 >𝑡𝑜

Hence there exists a y0 ∈ R y0 ∉ <µA>t0 but y0 ∈ T0(< 𝜇𝐴 >)𝑡𝑜. Thus

248 K. Meena

y0 =T0(x0) where x0 ∈ < 𝜇𝐴 >𝑡𝑜

⇒<µA>(x0) ≥ 𝑡𝑜

⇒ supr : x0 ∈ <(µA)r> ≥ t0. r∈Imµa

Since A possesses sup-property,

⇒ ∃ r0 ∈ Im 𝜇𝐴 є 𝑥𝑜 ∈ <(𝜇𝐴) 𝑟𝑜> and r0 ≥ t0 (14)

As 𝑥𝑜 ∈< (𝜇𝐴)𝑟𝑜> we get x0 = a0a1 ...an where ai ∈ (𝜇𝐴)𝑟𝑜

Hence T0𝑥𝑜 = 𝑇0(a0a1 ...an) = 𝑇0a0 · 𝑇0a1 ... 𝑇0an.

Also as µA ∈ CIFS (B), µA (𝑇0ai) ≥ µA (ai) and since ai ∈ (𝜇𝐴)𝑟0

⇒ 𝑇0𝑎𝑖 ∈(𝜇𝐴)𝑟0.

Consequently 𝑇0 𝑥𝑜∈< (𝜇𝐴)𝑟𝑜>. But y0 = T0(x0). Hence y0 ∈ < (𝜇𝐴)𝑟𝑜

>⇒

µA(y0) ≥ r0 ≥ t0 which is a contradiction.

Similarly, if <A> is not an CIFSR (B) ⇒ <A>∉ CIFSR (B) ⇒ ∃ t0 ∈ [0, 𝛾𝐵 (0)]

϶ < 𝜐𝐴 >𝑡0 is not a characteristic subring of (𝜐𝐵)𝑡0

.

⇒ ∃ 𝑇𝑜∈ A(𝛾𝐵)𝑡𝑜 є 𝑇𝑜 (< 𝛾𝐴 >𝑡𝑜

) ⊄ < 𝛾𝐴 >𝑡𝑜

Hence there exists y0 ∈ R ϶ y0 ∉ < 𝛾𝐴 >𝑡𝑜 but y0 ∈ T0 (< 𝛾𝐴 >𝑡𝑜)

Thus

y0 = T0(x0) where x0 ∈< 𝛾𝐴 >𝑡𝑜 ⇒<𝛾𝐴 >(x0) ≤ t0

.

Since A possesses sup-property ∃ r0 ∈ Im 𝛾𝐴 ϶ x0 ∈ <νA>r0 and r0 ≤ t0. As

x0 ∈ <νA>r0 we get

x0= b0b1b2 ...bn, where bi ∈(𝛾𝐴)𝑟𝑜.

Hence y0 = 𝑇0x0 = T0(b0b1b2 ...bn) = T0(b0)T0(b2)...T0(bn)

Also as A ∈ CIFS(B) implies νA(T0bi) ≤ νA(bi) and since bi ∈ (𝛾𝐴)𝑟𝑜 it follows that

T0bi ∈ (𝛾𝐴)𝑟𝑜 . Hence

𝑦𝑜 =𝑇0𝑥𝑜 ∈< 𝛾𝐴 >𝑟𝑜 ⇒ <𝛾𝐴>(𝑦𝑜) ≤ 𝑟𝑜 ≤ 𝑡𝑜


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 sub-

lattice 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 <𝑈𝑖Ai>

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).


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.

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 sub-lattice 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).


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.

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