Intuitionistic anti fuzzy normed ideals

In this paper, we introduce the notion of intuitionistic anti fuzzy normed subrings and intuitionistic anti fuzzy normed ideals and various properties of intuitionistic anti fuzzy normal subrings and ideals are examined. Also, we prove that the intersection of two intuitionistic anti fuzzy normed ideals is also an intuitionistic anti fuzzy normed ideal. Besides, we define the anti level subsets and identify the relationship between these sets and intuitionistic anti fuzzy normed ideals. In addition, we define the anti-image of intuitionistic anti fuzzy normed ideals and discuss the inverse image and anti-image for intuitionistic anti fuzzy normed ideals under epimorphism mapping.


Introduction
After the introduction of fuzzy sets by Zadeh [21] several researchers investigated on the generalization of the concept of fuzzy set. In 1971 [17], Rosenfeld initiated the studies of fuzzy group theory by introducing the concepts of fuzzy subgroupoid and fuzzy subgroup. Later in 1982, Liu [16] introduced the definition of fuzzy ring and discussed fuzzy subrings and fuzzy ideals and presented some basic concepts of fuzzy algebra, as fuzzy invariant subgroups, fuzzy ideals and proved some related properties. The notion of intuitionistic fuzzy set was introduced by Atanassov [5,6] as a generalization of fuzzy sets. After that many researches applied this notion in various branches of mathematics especially in algebra and defined the notions of intuitionistic fuzzy subgroups, intuitionistic fuzzy subrings and intuitionistic fuzzy normal subgroups. In 1989 [12], Biswas defined the notion of intuitionistic fuzzy groups and investigated its properties. Followed by a work in 2003 by Hur et al. in [14], which introduced the concepts of intuitionistic fuzzy subgroups and intuitionistic fuzzy subrings and examined some of their characteristics. In the same year, Banerjee and Basnet in [9], studied intuitionistic fuzzy subrings and ideals based on intuitionistic fuzzy sets. The notion of intuitionistic fuzzy normed rings was introduced by Abed Alhaleem and Ahmad in [1], in the paper they introduced the concepts of intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals as an extension of fuzzy normed rings which were studied by Emniyet and Sahin [13], which presented the notions of fuzzy normed subrings and fuzzy normed ideals. In 2011, Azam et al. [7] defined anti fuzzy ideal and lower level ideals of a ring and some related properties were proved. Sharma and Bansal [19] followed the work of Azam and introduced the notion of intuitionistic anti fuzzy subrings and intuitionistic anti fuzzy ideals and investigated various characteristics. In this paper, we introduce the notion of intuitionistic anti fuzzy normed subrings and intuitionistic anti fuzzy normed ideals. Further we characterize various properties related to intuitionistic fuzzy normed ideals and identify the anti image of intuitionistic anti fuzzy normed ideals.

Preliminaries
In this section, we state some basic definitions and results necessary for this paper.
Definition 2.1. [15] A normed ring NR is a ring that possesses a norm , that is a non-negative real-valued function : NR → R such that for all v, r ∈ NR, satisfies the following conditions: Let a set X be fixed. An intuitionistic fuzzy set A is an object having the form An intuitionistic fuzzy set A is written symbolically in the form A = (ϑ A , ζ A ) [8].
: v ∈ NR} is called an intuitionistic fuzzy normed subring (IFNSR) of the normed ring (NR, +, .) if it satisfies the following conditions for all v, r ∈ NR:

Definition 2.4. [1]
If the fuzzy set A = (ϑ A , ζ A ) is both right and left intuitionistic fuzzy normed ideal of NR, then A is called intuitionistic fuzzy normed ideal (IFNI), if for every v, r ∈ NR: Proposition 2.5.
[1] Let A be an intuitionistic fuzzy normed ring and let 0 be the zero of the normed ring NR, then the following is true for every v ∈ NR: Lemma 2.6. [1] Let 1 NR be the multiplicative identity of NR then for all v ∈ NR: Theorem 2.9.
[10] Let f : R → R be a surjective ring homomorphism, then

Intuitionistic anti fuzzy normed ideals
In this section, we introduce the notions of intuitionistic anti fuzzy normed subrings and intuitionistic anti fuzzy normed ideals. Further, we generalize some basic related properties.
be an intuitionistic anti fuzzy normed subring of NR, then for every v, r ∈ NR: ii. Similar to the proof of i.

Remark 3.3.
If A is an intuitionistic anti fuzzy normed subring of NR, then: and ϑ A (r) at least two are equal, and and ζ A (r) at least two are equal. Proof. Suppose that A is an IAFNSR, we need to show that iii.
. On the other hand, we can prove similarly that if A c is an intuitionistic fuzzy normed subring, then A is an IAFNSR of NR.
Definition 3.5. An intuitionistic fuzzy set A of a normed ring NR is said to be an intuitionistic anti fuzzy normed left (right) ideal of NR if for all v, r ∈ NR: Definition 3.6. An IFS A of a normed ring NR is said to be an intuitionistic anti fuzzy normed ideal (IAFNI) of NR if it satisfies the following conditions for all v, r ∈ NR: Proof. Suppose that A is an IAFNI, we need to show that A c = (ϑ A c , ζ A c ) = (ζ A , ϑ A ) is an intuitionistic fuzzy normed ideal of NR. i.
. On the other hand, we can prove similarly that if A c is an intuitionistic fuzzy normed ideal, then A is an IAFNI of NR. Proof. Let v, r ∈ NR. Then, i.
Hence, A ∩ B is an intuitionistic anti fuzzy normed left ideal. Likewise it can be proven that A ∩ B is an intuitionistic anti fuzzy normed right ideal. So, A ∩ B is an IAFNI of NR.
Remark 3.9. The union of two intuitionistic anti fuzzy normed ideals of a ring NR needs not be always intuitionistic anti fuzzy normed ideal.
Example 3.1. Let NR = Z be the ring of integers under ordinary addition and multiplication of integers.Let the intuitionistic anti fuzzy normed subsets A = (ϑ A , ζ A ) and B = (ϑ B , ζ B ), be defined by It can be checked that A and B are IAFNI of Z.
Proposition 3.10. Let A be an intuitionistic anti fuzzy normed ideal of a ring NR, then we have for all v ∈ NR: Proof. i. As A is an IAFNI, then and Similarly, Let z ∈ NR and v ∈ A * . We have, This implies that, ϑ A (zv) = ϑ A (0) and ζ A (zv) = ζ A (0). Therefore, zv ∈ A * . Thus, A * is a left ideal of NR.
Theorem 3.13. Let A be an intuitionistic fuzzy set in a ring NR. If A is an intuitionistic anti fuzzy left (right) ideal of NR, then each anti level subsetĀ α,β , is a left (right) ideal of NR.
is an intuitionistic fuzzy normed subring in NR such that f - Theorem 3.17. A homomorphic inverse image of an intuitionistic anti fuzzy normed ideal is an intuitionistic anti fuzzy normed ideal.
Proof. Let f : NR → NR be a ring homomorphism. Let B be an IAFNI in NR and f −1 be the inverse image of B. Suppose v, r ∈ NR, then we have: i. ii. iii. iv.
. ii. If A is an intuitionistic fuzzy set in NR, then f (A c ) = ( f − (A)) c .
Proof. i. As B is an intuitionistic fuzzy set in NR , then we have for every v ∈ NR: ii. As A is an intuitionistic fuzzy set in NR, then we have for every r ∈ NR : ii. If A is an intuitionistic anti fuzzy normed ideal of NR, then f − (A) is an intuitionistic anti fuzzy normed ideal of NR .
Proof. i. As B is an IAFNI in NR , then B c is an intuitionistic fuzzy normed ideal in NR and then f -1 (B c ) is an intuitionistic fuzzy normed ideal in NR which means that ( f -1 (B)) c is an intuitionistic fuzzy normed ideal in NR. Hence, f -1 (B) is an IAFNI in NR.
ii. As A is an IAFNI in NR, then A c is an intuitionistic fuzzy normed ideal in NR and then f (A c ) is an intuitionistic fuzzy normed ideal in NR . As f (A c ) = ( f − (A)) c which means that ( f − (A)) c is an intuitionistic fuzzy normed ideal in NR . Hence, f − (A) is an IAFNI in NR .