A two-step randomized Gauss-Seidel (TRGS) method is presented for large linear least squares problem with tall and narrow coefficient matrix. The TRGS method projects the approximate solution onto the solution space by given two random columns and is proved to be convergent when the coefficient matrix is of full rank. Several numerical examples show the effectiveness of the TRGS method among all methods compared.
Citation: Yimou Liao, Tianxiu Lu, Feng Yin. A two-step randomized Gauss-Seidel method for solving large-scale linear least squares problems[J]. Electronic Research Archive, 2022, 30(2): 755-779. doi: 10.3934/era.2022040
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A two-step randomized Gauss-Seidel (TRGS) method is presented for large linear least squares problem with tall and narrow coefficient matrix. The TRGS method projects the approximate solution onto the solution space by given two random columns and is proved to be convergent when the coefficient matrix is of full rank. Several numerical examples show the effectiveness of the TRGS method among all methods compared.
The notion of intuitionistic fuzzy normed subring and intuitionistic fuzzy normed ideal was characterized by Abed Alhaleem and Ahmad in [10], after that the necessity has arisen to introduce the concepts of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals. Following the work of Emniyent and Şahin in [17] which outlined the concepts of fuzzy normed prime ideal and maximal ideal we implement the conception of intuitionistic fuzzy to prime and maximal normed ideals. After the establishment of fuzzy set by Zadeh [28] which showed that the membership of an element in a fuzzy set is at intervals [0, 1], many researchers investigated on the properties of fuzzy set because it handles uncertainty and vagueness, and due to its applications in many fields of studies. A lot of work has been done on various aspects and for the last 50 years, the relation betwee maximal and prime ideals has become the core of many researchers work. Swamy and Swamy in 1988 [27] presented the conceptions of fuzzy ideal and fuzzy prime ideal with truth values in a complete lattice fulfilling the infinite distributive law. Later, many researchers studied the generalization of fuzzy ideals and fuzzy prime (maximal) ideals of rings: Dixit et al [16], Malik and Mordeson in [22] and Mukherjee and Sen in [24]. The notion of intuitionistic fuzzy set was initiated by Atanassov [6], as a characterization of fuzzy set which assigned the degree of membership and the degree of non-membership for set elements, he also delineated some operations and connections over basic intuitionistic fuzzy sets. In [5], Atanassov introduced essential definitions and properties of the interval-valued intuitionistic fuzzy sets and the explanation of mostly extended modal operator through interval-valued intuitionistic fuzzy sets were presented in [4], and some of its main properties were studied. Banerjee and Basnet [13] investigated intuitionistic fuzzy rings and intuitionistic fuzzy ideals using intuitionistic fuzzy sets. In 2005 [20], an identification of intuitionistic fuzzy ideals, intuitionistic fuzzy prime ideals and intuitionistic fuzzy completely prime ideals was given. In [14], Bakhadach et al. implemented the terms of intuitionistic fuzzy ideals and intuitionistic fuzzy prime (maximal) ideals, investigated these notions to show new results using the intuitionistic fuzzy points and membership and nonmembership functions. The paper comprises the following: we begin with the preliminary section, we submit necessary notations and elementary outcomes. In Section 3, we characterize some properties of intuitionistic fuzzy normed ideals and identify the image and the inverse image of intuitionistic fuzzy normed ideals. In Section 4, we describe the notions of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals and we characterize the relation between the intuitionistic characteristic function and prime (maximal) ideals. In Section 5, the conclusions are outlined.
We first include some definitions needed for the subsequent sections:
Definition 2.1. [25] A linear space L is called a normed space if for any element r there is a real number ‖r‖ satisfying:
∙‖r‖≥0 for every r∈L, when r=0 then ‖r‖=0;
∙‖α.r‖=|α|.‖r‖;
∙‖r+v‖≤‖r‖+‖v‖ for all r,v∈L.
Definition 2.2. [18] A ring R is said to be a normed ring (NR) if it possesses a norm ‖‖, that is, a non-negative real-valued function ‖‖:NR→R such that for any r,v∈R,
1)‖r‖=0⇔r=0,
2)‖r+v‖≤‖r‖+‖v‖,
3)‖r‖=‖−r‖, (and hence ‖1A‖=1=‖−1‖ if identity exists), and
4)‖rv‖≤‖r‖‖v‖.
Definition 2.3. [1] Let ∗:[0,1]×[0,1]→[0,1] be a binary operation. Then ∗ is a t-norm if ∗ conciliates the conditions of commutativity, associativity, monotonicity and neutral element 1.
We shortly use t-norm and write r∗v instead of ∗(r,v).
Two examples of continuous t-norm are: r∗v=rv and r∗v=min{r,v} [26].
Proposition 2.4. [21] A t-norm T has the property, for every r,v∈[0,1]
T(r,v)≤min(r,v) |
Definition 2.5. [19] Let ⋄:[0,1]×[0,1]→[0,1] be a binary operation. Then ⋄ is a s-norm if ⋄ conciliates the conditions of commutativity, associativity, monotonicity and neutral element 0.
We shortly use s-norm and write r⋄v instead of ⋄(r,v).
Two examples of continuous s-norm are: r⋄v=min(r+v,1) and r⋄v=max{r,v} [26].
Proposition 2.6. [21] A s-norm S has the property, for every r,v∈[0,1]
max(r,v)≤S(r,v) |
Definition 2.7. [28] A membership function μA(r):X→[0,1] specifies the fuzzy set A over X, where μA(r) defines the membership of an element r∈X in a fuzzy set A.
Definition 2.8. [6] An intuitionistic fuzzy set A in set X is in the form IFSA={(r,μA(r),γA(r):r∈X}, such that the degree of membership is μA(r):X→[0,1] and the degree of non-membership is γA(r):X→[0,1], where 0≤μA(r)+γA(r))≤1 for all r∈X. We shortly use A=(μA,γA).
Definition 2.9. [7] Let A be an intuitionistic fuzzy set in a ring R, we indicate the (α,β)-cut set by Aα,β={r∈R:μA≥α and γA≤β} such that α+β≤1 and α,β∈[0,1].
Definition 2.10. [23] The support of an intuitionistic fuzzy set A, is denoted by A∘ and defined as A∘={r:μA(r)>0 and γA(r)<1}.
Definition 2.11. [2] The complement, union and intersection of two IFSA=(μA,γA) and B=(μB,γB), in a ring R, are defined as follows:
1)Ac={⟨r,γA(r),μA(r)⟩:r∈R},
2)A∪B={⟨r,max(μA(r),μB(r)),min(γA(r),γB(r))⟩:r∈R},
3)A∩B={⟨r,min(μA(r),μB(r)),max(γA(r),γB(r))⟩:r∈R}.
Definition 2.12. [12] Let NR be a normed ring. Then an IFS A={(r,μA(r),γA(r)):r∈NR} of NR is an intuitionistic fuzzy normed subring (IFNSR) of NR if:
i. μA(r−v)≥μA(r)∗μA(v),
ii. μA(rv)≥μA(r)∗μA(v),
iii. γA(r−v)≤γA(r)⋄γA(v),
iv. γA(rv)≤γA(r)⋄γA(v).
Definition 2.13. [9] Let NR be a normed ring. Then an IFS A={(r,μA(r),γA(r)):r∈NR} of NR is an intuitionistic fuzzy normed ideal (IFNI) of NR if:
i. μA(r−v)≥μA(r)∗μA(v),
ii. μA(rv)≥μA(r)⋄μA(v),
iii. γA(r−v)≤γA(r)⋄γA(v)),
iv. γA(rv))≤γA(r)∗γA(v)}.
Definition 2.14. [3] If A and B are two fuzzy subsets of the normed ring NR. Then the product A∘B(r) is defined by:
A∘B(r)={⋄r=vz(μA(v)∗μB(z)),ifr=vz0,otherwise |
Definition 2.15. [22] A fuzzy ideal A (non-constant) of a ring R is considered to be a fuzzy prime ideal if B∘C⊆A for a fuzzy ideals B, C of R indicates that either B⊆A or C⊆A.
In this section, we characterize several properties of intuitionistic fuzzy normed ideals and elementary results are obtained.
Definition 3.1. [8] Let A and B be two intuitionistic fuzzy subsets of the normed ring NR. The operations are defined as:
μA⊛B(r)={⋄r=vz(μA(v)∗μB(z)),ifr=vz0,otherwise |
and
γA⊗B(r)={∗r=vz(γA(v)⋄γB(z)),ifr=vz1,otherwise |
Therefore, the intrinsic product of A and B is considered to be the intuitionistic fuzzy normed set A∘B=(μA⊛B,γA⊗B)=(μA⊛μB,γA⊗γB).
Theorem 3.2. [10] Let A and B be two intuitionistic fuzzy ideals of a normed ring NR. Then A∩B is an intuitionistic fuzzy normed ideal of NR.
Example 3.1. Let NR=Z the ring of integers under ordinary addition and multiplication of integers.
Define the intuitionistic fuzzy normed subsets as A=(μA,γA) and B=(μB,γB), by
μA(r)={0.7,ifr∈5Z0.2,otherwiseandγA(r)={0.1,ifr∈5Z0.4,otherwise |
μB(r)={0.8,ifr∈5Z0.3,otherwiseandγB(r)={0.2,ifr∈5Z0.7,otherwise |
As μA∩B(r)=min{μA(r),μB(r)} and γA∩B(r)=max{γA(r),γB(r)}. Then,
μA∩B(r)={0.7,ifr∈5Z0.2,otherwiseandγA∩B(r)={0.2,ifr∈5Z0.7,otherwise |
It can be verified that A, B and A∩B are intuitionistic fuzzy normed ideals of NR.
Lemma 3.3. Let A and B be an intuitionistic fuzzy normed right ideal and an intuitionistic fuzzy normed left ideal of a normed ring NR, respectively, then A∘B⊆A∩B i.e, A⊛B(r)≤A∩B(r)≤A⊗B(r), where
A∩B(r)={(r,μA∩B(r),γA∩B(r)):r∈NR}={(r,min{μA(r),μB(r)},max{γA(r),γB(r)}):r∈NR}. |
Proof. Let A∩B be an intuitionistic fuzzy normed ideal of NR. Assume that A is an intuitionistic fuzzy normed right ideal and B is an intuitionistic fuzzy normed left ideal. Let μA⊛B(r)=⋄r=vz(μA(v)∗μB(z)) and let γA⊗B(r)=∗r=vz(γA(v)⋄γB(z)).
Since, A is an intuitionistic fuzzy normed right ideal and B is an intuitionistic fuzzy normed left ideal, we have
μA(v)≤μA(vz)=μA(r)andμB(z)≤μB(vz)=μB(r) |
and
γA(r)=γA(vz)≥γA(v)andγB(r)=γB(vz)≥γB(z). |
Thus,
μA⊛B(r)=⋄r=vz(μA(v)∗μB(z))=min(μA(v),μB(z))≤min(μA(r),μB(r))≤μA∩B(r) | (3.1) |
and
γA⊗B(r)=∗r=vz(γA(v)⋄γB(z))=max(γA(v),γB(z))≥max(γA(r),γB(r))≥γA∩B(r). | (3.2) |
By (3.1) and (3.2) the proof is concluded.
Remark 3.4. The union of two intuitionistic fuzzy normed ideals of a ring NR needs not be always intuitionistic fuzzy normed ideal.
Example 3.2. Let NR=Z the ring of integers under ordinary addition and multiplication of integers.
Let the intuitionistic fuzzy normed subsets A=(μA,γA) and B=(μB,γB), define by
μA(r)={0.85,ifr∈3Z0.3,otherwiseandγA(r)={0.2,ifr∈3Z0.4,otherwise |
μB(r)={0.75,ifr∈2Z0.35,otherwiseandγB(r)={0.3,ifr∈2Z0.5,otherwise |
It can be checked that A and B are intuitionistic fuzzy normed ideals of NR.
As μA∪B(r)=max{μA(r),μB(r)} and γA∪B(r)=min{γA(r),γB(r)}. Then,
μA∪B(r)={0.85,ifr∈3Z0.75,ifr∈2Z−3Z0.35,ifr∉2Zorr∉3ZandγA∪B(r)={0.2,ifr∈3Z0.3,ifr∈2Z−3Z0.4,ifr∉2Zorr∉3Z |
Let r=15 and v=4, then μA∪B(15)=0.85, μA∪B(4)=0.75 and γA∪B(15)=0.2, γA∪B(4)=0.3.
Hence, μA∪B(15−4)=μA∪B(11)=0.35≯μA∪B(15)∗μA∪B(4)=min{0.85,0.75} and γA∪B(15−4)=γA∪B(11)=0.4≮γA∪B(15)⋄γA∪B(4)=max{0.2,0.3}. Thus, the union of two intuitionistic fuzzy normed ideals of NR need not be an intuitionistic fuzzy normed ideal.
Proposition 3.5. Let A=(μA,γA) be an intuitionistic fuzzy normed ideal of a ring NR, then we have for all r∈NR:
i. μA(0)≥μA(r) and γA(0)≤γA(r),
ii. μA(−r)=μA(r) and γA(−r)=γA(r),
iii. If μA(r−v)=μA(0) then μA(r)=μA(v),
iv. If γA(r−v)=γA(0) then γA(r)=γA(v).
Proof. i. As A is an intuitionistic fuzzy normed ideal, then
μA(0)=μA(r−r)≥μA(r)∗μA(r)=μA(r) |
and
γA(0)=γA(r−r)≤γA(r)⋄γA(r)=γA(r) |
ii. μA(−r)=μA(0−r)≥μA(0)∗μA(r)=μA(r) and μA(r)=μA(0−(−r))≥μA(0)∗μA(−r)=μA(−r).
Therefore, μA(−r)=μA(r)
also,
γA(−r)=γA(0−r)≤γA(0)⋄γA(r)=γA(r) and γA(r)=γA(0−(−r))≤γA(0)⋄γA(−r)=γA(−r).
Therefore, γA(−r)=γA(r).
iii. Since μA(r−v)=μA(0), then
μA(v)=μA(r−(r−v))≥μA(r)∗μA(r−v)=μA(r)∗μA(0)≥μA(r) |
similarly
μA(r)=μA((r−v)−(−v))≥μA(r−v)∗μA(−v)=μA(0)∗μA(v)≥μA(v) |
Consequently, μA(r)=μA(v).
iv. same as in iii.
Proposition 3.6. Let A be an intuitionistic fuzzy normed ideal of a normed ring NR, then △A=(μA,μcA) is an intuitionistic fuzzy normed ideal of NR.
Proof. Let r,v∈NR
μcA(r−v)=1−μA(r−v)≤1−min{μA(r),μA(v)}=max{1−μA(r),1−μA(v)}=max{μcA(r),μcA(v)} |
Then μcA(r−v)≤μcA(r)⋄μcA(v).
μcA(rv)=1−μA(rv)≤1−max{μA(r),μA(v)}=min{1−μA(r),1−μA(v)}=min{μcA(r),μcA(v)} |
Then μcA(rv)≤μcA(r)∗μcA(v).
Accordingly, △A=(μA,μcA) is an intuitionistic fuzzy normed ideal of NR.
Proposition 3.7. If A is an intuitionistic fuzzy normed ideal of a normed ring NR, then ◊A=(γcA,γA) is an intuitionistic fuzzy normed ideal of NR.
Proof. Let r,v∈NR
γcA(r−v)=1−γA(r−v)≥1−max{γA(r),γA(v)}=min{1−γA(r),1−γA(v)}=min{γcA(r),γcA(v)} |
Then γcA(r−v)≥γcA(r)∗γcA(v).
γAc(rv)=1−γA(rv)≥1−min{γA(r),γA(v)}=max{1−μA(r),1−γA(v)}=max{γcA(r),γcA(v)} |
Then γcA(rv)≥γcA(r)⋄γcA(v).
Therefore, ◊A=(γcA,γA) is an intuitionistic fuzzy normed ideal of NR.
Proposition 3.8. An IFSA=(μA,γA) is an intuitionistic fuzzy normed ideal of NR if the fuzzy subsets μA and γcA are intuitionistic fuzzy normed ideals of NR.
Proof. Let r,v∈NR
1−γA(r−v)=γcA(r−v)≥min{γcA(r),γcA(v)}=min{(1−γA(r)),(1−γA(v))}=1−max{γA(r),γA(v)} |
Then, γA(r−v)≤γA(r)⋄γA(v).
1−γA(rv)=γcA(rv)≥max{γcA(r),γcA(v)}=max{(1−γA(r)),(1−γA(v))}=1−min{γA(r),γA(v)} |
Then, γA(rv)≤γA(r)∗γA(v).
Consequently, A=(μA,γA) is an intuitionistic fuzzy normed ideal of NR.
Definition 3.9. Let A be a set (non-empty) of the normed ring NR, the intuitionistic characteristic function of A is defined as λA=(μλA,γλA), where
μλA(r)={1,ifr∈A0,ifr∉AandγλA(r)={0,ifr∈A1,ifr∉A |
Lemma 3.10. Let A and B be intuitionistic fuzzy sets of a normed ring NR, then:
(i) λA∩λB=λA∩B (ii) λA∘λB=λA∘B (iii) If A⊆B, then λA⊆λB
Theorem 3.11. For a non-empty subset A of NR, A is a subring of NR if and only if λA=(μλA,γλA) is an intuitionistic fuzzy normed subring of NR.
Proof. Suppose A to be a subring of NR and let r,v∈NR. If r,v∈A, then by the intuitionistic characteristic function properties μλA(r)=1=μλA(v) and γλA(r)=0=γλA(v). As A is a subring, then r−v and rv∈A. Thus, μλA(r−v)=1=1∗1=μλA(r)∗μλA(v) and μλA(rv)=1=1∗1=μλA(r)∗μλA(v), also γλA(r−v)=0=0⋄0=γλA(r)⋄γλA(v) and γλA(rv)=0=0⋄0=γλA(r)⋄γλA(v). This implies,
μλA(r−v)≥μλA(r)∗μλA(v)andμλA(rv)≥μλA(r)∗μλA(v),γλA(r−v)≤γλA(r)⋄γλA(v)andγλA(rv)≤γλA(r)⋄γλA(v). |
Similarly we can prove the above expressions if r,v∉A.
Hence, λA=(μλA,γλA) is an intuitionistic fuzzy normed subring of NR.
Conversely, we hypothesise that the intuitionistic characteristic function λA=(μλA,γλA) is an intuitionistic fuzzy normed subring of NR. Let r,v∈A, then μλA(r)=1=μλA(v) and γλA(r)=0=γλA(v). So,
μλA(r−v)≥μλA(r)∗μλA(v)≥1∗1≥1,alsoμλA(r−v)≤1,μλA(rv)≥μλA(r)∗μλA(v)≥1∗1≥1,alsoμλA(rv)≤1,γλA(r−v)≤γλA(r)⋄γλA(v)≤0⋄0≤0,alsoγλA(r−v)≥0,γλA(rv)≤γλA(r)⋄γλA(v)≤0⋄0≤0,alsoγλA(rv)≥0, |
then μλA(r−v)=1, μλA(rv)=1 and γλA(r−v)=0, γλA(rv)=0, which implies that r−v and rv∈A. Therefore, A is a subring of NR.
Theorem 3.12. Let I be a non-empty subset of a normed ring NR, then I is an ideal of NR if and only if λI=(μλI,γλI) is an intuitionistic fuzzy normed ideal of NR.
Proof. Let I be an ideal of NR and let r,v∈NR.
Case I. If r,v∈I then rv∈I and μλI(r)=1, μλI(v)=1 and γλI(r)=0, γλI(v)=0. Thus, μλI(rv)=1 and γλI(rv)=0. Accordingly, μλI(rv)=1=μλI(r)⋄μλI(v) and γλI(rv)=0=γλI(r)∗γλI(v).
Case II. If r∉I or v∉I so rv∉I, then μλI(r)=0 or μλI(v)=0 and γλI(r)=1 or γλI(v)=1. So, μλI(rv)=1≥μλI(r)⋄μλI(v) and γλI(rv)=0≤γλI(r)∗γλI(v). Hence, λI=(μλI,γλI) is an intuitionistic fuzzy normed ideal of NR.
On the hand, we suppose λI=(μλI,γλI) is an intuitionistic fuzzy normed ideal of NR. The proof is similar to the second part of the proof of Theorem 3.11.
Proposition 3.13. If A is an intuitionistic fuzzy normed ideal of NR, then A∗ is an ideal of NR where A∗ is defined as,
A∗={r∈NR:μA(r)=μA(0)andγA(r)=γA(0)} |
Proof. See [10] (p. 6)
Lemma 3.14. Let A and B be two intuitionistic fuzzy normed left (right) ideal of NR. Therefore, A∗∩B∗⊆(A∩B)∗.
Proof. Let r∈A∗∩B∗, then μA(r)=μA(0), μB(r)=μB(0) and γA(r)=γA(0), γB(r)=γB(0).
μA∩B(r)=min{μA(r),μB(r)}=min{μA(0),μB(0)}=μA∩B(0) |
and
γA∩B(r)=max{γA(r),γB(r)}=max{γA(0),γB(0)}=γA∩B(0) |
So, r∈(A∩B)∗. Thus, A∗∩B∗⊆(A∩B)∗.
Theorem 3.15. Let f:NR→NR′ be an epimorphism mapping of normed rings. If A is an intuitionistic fuzzy normed ideal of the normed ring NR, then f(A) is also an intuitionistic fuzzy normed ideal of NR′.
Proof. Suppose A={(r,μA(r),γA(r)):r∈NR},
f(A)={(v,⋄f(r)=vμA(r),∗f(r)=vγA(r):r∈NR,v∈NR′}.
Let v1,v2∈NR′, then there exists r1,r2∈NR such that f(r1)=v1 and f(r2)=v2.
i.
μf(A)(v1−v2)=⋄f(r1−r2)=v1−v2μA(r1−r2)≥⋄f(r1)=v1,f(r2)=v2(μA(r1)∗μA(r2))≥(⋄f(r1)=v1μA(r1))∗(⋄f(r2)=v2μA(r2))≥μf(A)(v1)∗μf(A)(v2) |
ii.
μf(A)(v1v2)=⋄f(r1r2)=v1v2μA(r1r2)≥⋄f(r2)=v2μA(r2)≥μf(A)(v2) |
iii.
γf(A)(v1−v2)=∗f(r1−r2)=v1−v2γA(r1−r2)≤∗f(r1)=v1,f(r2)=v2(γA(r1)⋄γA(r2))≤(∗f(r1)=v1γA(r1))⋄(∗f(r2)=v2γA(r2))≤γf(A)(v1)⋄γf(A)(v2) |
iv.
γf(A)(v1v2)=∗f(r1r2)=v1v2γA(r1r2)≤∗f(r2)=v2γA(r2)≤γf(A)(v2) |
Hence, f(A) is an intuitionistic fuzzy normed left ideal. Similarly, it can be justified that f(A) is an intuitionistic fuzzy normed right ideal. Then, f(A) is a intuitionistic fuzzy normed ideal of NR′.
Proposition 3.16. Define f:NR→NR′ to be an epimorphism mapping. If B is an intuitionistic fuzzy normed ideal of the normed ring NR′, then f−1(B) is also an intuitionistic fuzzy normed ideal of NR.
Proof. Suppose B={(v,μB(v),γB(v)):v∈NR′}, f−1(B)={(r,μf−1(B)(r),γf−1(B)(r):r∈NR}, where μf−1(B)(r)=μB(f(r)) and γf−1(B)(r)=γB(f(r)) for every r∈NR. Let r1,r2∈NR, then
i.
μf−1(B)(r1−r2)=μB(f(r1−r2))=μB(f(r1)−f(r2))≥μB(f(r1))∗μB(f(r2))≥μf−1(B)(r1)∗μf−1(B)(r2) |
ii.
μf−1(B)(r1r2)=μB(f(r1r2))=μB(f(r1)f(r2))≥μB(f(r2))≥μf−1(B)(r2) |
iii.
γf−1(B)(r1−r2)=γB(f(r1−r2))=γB(f(r1)−f(r2))≤γB(f(r1))⋄γB(f(r2))≤γf−1(B)(r1)⋄γf−1(B)(r2) |
iv.
γf−1(B)(r1r2)=γB(f(r1r2))=γB(f(r1)f(r2))≤γB(f(r2))≤γf−1(B)(r2) |
Therefore, f−1(B) is an intuitionistic fuzzy normed left ideal of NR. Similarly, it can be justified that f−1(B) is an intuitionistic fuzzy normed right ideal. So, f−1(B) is a intuitionistic fuzzy normed ideal of NR.
In what follows, we produce the terms of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals and we investigate some associated properties.
Definition 4.1. An intuitionistic fuzzy normed ideal A=(μA,γA) of a normed ring NR is said to be an intuitionistic fuzzy normed prime ideal of NR if for an intuitionistic fuzzy normed ideals B=(μB,γB) and C=(μC,γC) of NR where B∘C⊆A indicates that either B⊆A or C⊆A, which imply that μB⊆μA and γA⊆γB or μC⊆μA and γA⊆γC.
Proposition 4.2. An intuitionistic fuzzy normed ideal A=(μA,γA) is an intuitionistic fuzzy normed prime ideal if for any two intuitionistic fuzzy normed ideals B=(μB,γB) and C=(μC,γC) of NR satisfies:
i. μA⊇μB⊛C i.e. μA(r)≥⋄r=vz(μA(v)∗μB(z));
ii. γA⊆γB⊗C i.e.γA(r)≤∗r=vz(γA(v)⋄γB(z)).
Theorem 4.3. Let A be an intuitionistic fuzzy normed prime ideal of NR. Then ∣Im μA∣ = ∣Im γA∣=2; in other words A is two-valued.
Proof. As A is not constant, ∣Im μA∣≥2. assume that ∣Im μA∣≥3. Aα,β={r∈R:μA≥α and γA≤β} where α+β≤1. Let r∈NR and let B and C be two intuitionistic fuzzy subsets in NR, such that: μA(0)=s and k=glb{μA(r):r∈NR}, so there exists t,α∈ Im(μA) such that k≤t<α<s with μB(r)=12(t+α), μC(r)={s,ifr∈Aα,βk,ifr∉Aα,β and γA(0)=c and h=lub{γA(r):r∈NR}, then there exists d,β∈ Im(γA) such that c<β<d≤h with γB(r)=12(d+β) and γC(r)={c,ifr∈Aα,βh,ifr∉Aα,β for all r∈NR. Clearly B is an intuitionistic fuzzy normed ideal of NR. Now we claim that C is an intuitionistic fuzzy normed ideal of NR.
Let r,v∈NR, if r,v∈Aα,β then r−v∈Aα,β and μC(r−v)=s=μC(r)∗μC(v), γC(r−v)=c=γC(r)⋄γC(v). If r∈Aα,β and v∉Aα,β then r−v∉Aα,β so, μC(r−v)=k=μC(r)∗μC(v), γC(r−v)=h=γC(r)⋄γC(v). If r,v∉Aα,β then r−v∉Aα,β so, μC(r−v)≥k=μC(r)∗μC(v), γC(r−v)≤h=γC(r)⋄γC(v). Hence, μC(r−v)≥μC(r)∗μC(v) and γC(r−v)≤γC(r)⋄γC(v) for all r,v∈NR.
Now if r∈Aα,β then rv∈Aα,β, thus μC(rv)=s=μC(r)⋄μC(v) and γC(rv)=c=γC(r)∗γC(v). If r∉Aα,β, then μC(rv)≥k=μC(r)⋄μC(v) and γC(rv)≤h=γC(r)∗γC(v). Therefore C is an intuitionistic fuzzy normed ideal of NR.
To prove that B∘C⊆A. Let r∈NR, we discuss the following cases:
(i) If r=0, consequently
μB⊛C(0)=⋄r=uv(μB(u)∗μC(v))≤12(t+α)<s=μA(0); |
γB⊗C(r)=∗r=uv(γB(u)⋄γC(v))≥12(d+β)>c=γA(0). |
(ii) If r≠0, r∈Aα,β. Then μA(r)≥α and γA(r)≤β. Thus,
μB⊛C(r)=⋄r=uv(μB(u)∗μC(v))≤12(t+α)<α≤μA(r); |
γB⊗C(r)=∗r=uv(γB(u)⋄γC(v))≥12(d+β)>β≥γA(r). |
Since μB(u)∗μC(v)≤μB(u) and γB(u)⋄γC(v)≥γB(u).
(iii) If r≠0, r∉Aα,β. Then in that case u,v∈NR such that r=uv, u∉Aα,β and v∉Aα,β. Then,
μB⊛C(r)=⋄r=uv(μB(u)∗μC(v))=k≤μA(r); |
γB⊗C(r)=∗r=uv(γB(u)⋄γC(v))=h≥γA(r). |
Therefore, in any case μB⊛C(r)≤μA(r) and γB⊗C(r)≥γA(r) for all r∈NR. Hence, B∘C⊆A.
Let a,b∈NR such that μA(a)=t, μA(b)=α and γA(a)=d, γA(b)=β. Thus, μB(a)=12(t+α)>t=μA(r) and γB(a)=12(d+β)<d=γA(r) which implies that B⊈A. Also, μA(b)=α and γA(b)=β imply that b∈Aα,β so, μC(b)=s>α and γC(b)=c<β, so C⊈A. Therefore, neither B⊈A nor C⊈A. This indicates that A could not be an intuitionistic fuzzy normed prime ideal of NR, so its a contradiction. Thus, ∣Im μA∣ = ∣ImγA∣=2.
Proposition 4.4. If A is an intuitionistic fuzzy normed prime ideal of NR, so the following are satisfied:
i. μA(0NR)=1 and γA(0NR)=0;
ii. Im(μA)={1,α} and Im(γA)={0,β}, where α,β∈[0,1];
iii. A∗ is a prime ideal of NR.
Theorem 4.5. Let A be a fuzzy subset of NR where A is two-valued, μA(0)=1 and γA(0)=0, and the set A∗={r∈NR:μA(r)=μA(0) and γA(r)=γA(0)} is a prime ideal of NR. Hence, A is an intuitionistic fuzzy normed prime ideal of NR.
Proof. We have Im(μA)={1,α} and Im(γA)={0,β}. Let r,v∈NR. If r,v∈A∗, then r−v∈A∗ so, μA(r−v)=1=μA(r)∗μA(v) and γA(r−v)=0=γA(r)⋄γA(v). If r,v∉A∗, then μA(r−v)=α≥μA(r)∗μA(v) and γA(r−v)=β≤γA(r)⋄γA(v).
Therefore, for all r,v∈NR,
μA(r−v)≥μA(r)∗μA(v)γA(r−v)≤γA(r)⋄γA(v) |
Similarly,
μA(rv)≥μA(r)⋄μA(v)γA(rv)≤γA(r)∗γA(v) |
Thus A is an intuitionistic fuzzy ideal of NR.
Assume B and C be fuzzy ideals of NR where B∘C⊆A. Assume that B⊈A and C⊈A. Then, we have r,v∈NR in such a way that μB(r)>μA(r) and γB(r)<γA(r), μC(v)>μA(v) and γC(r)<γA(r), so for all a∈A∗, μA(a)=1=μA(0) and γA(a)=0=γA(0), r∉A∗ and v∉A∗. Since, A∗ is a prime ideal of NR, we have n∈NR in such a way that rnv∉A∗. Let a=rnv then μA(a)=μA(r)=μA(v)=α and γA(a)=γA(r)=γA(v)=β, now
μB⊛C(a)=⋄a=st(μB(s)∗μC(t))≥μB(r)∗μC(nv)≥μB(r)∗μC(v)>α=μA(a)[Since,μB(r)≥μA(r)=αandμC(nv)≥μC(v)≥μA(v)=α]. |
and
γB⊗C(a)=∗a=st(γB(s)⋄γC(t))≤γB(r)⋄γC(nv)≤γB(r)⋄γC(v)<β=γA(a)[Since,γB(r)≤γA(r)=βandγC(nv)≤γC(v)≤γA(v)=β]. |
Which means that B∘C⊈A. Which contradicts with the hypothesis that B∘C⊆A. Therefore, either B⊆A or C⊆A. Then A is an intuitionistic fuzzy normed prime ideal.
Theorem 4.6. Let P be a subset (non-empty) of NR. P is a prime ideal if and only if the intuitionistic characteristic function λP=(μλP,γλP) is an intuitionistic fuzzy normed prime ideal.
Proof. presume that P is a prime ideal of NR. So by Theorem 3.12, λP is an intuitionistic fuzzy normed ideal of NR. Let A=(μA,γA) and B=(μB,γB) be any intuitionistic fuzzy normed ideals of NR with A∘B⊆λP while A⊈λP and B⊈λP. Then there exist r,v∈NR such that
μA(r)≠0,γA(r)≠1andμB(v)≠0,γB(v)≠1 |
but
μλP(r)=0,γλP(r)=1andμλP(v)=0,γλP(v)=1 |
Therefore, r∉P and v∉P. Since P is a prime ideal, there exist n∈NR such that rnv∉P.
Let a=rnv, then μλP(a)=0 and γλP(a)=1. Thus, μA⊛b(a)=0 and γA⊗B(a)=1. but
μA⊛B(a)=⋄a=st(μA(s)∗μB(t))≥μA(r)∗μB(nv)≥μA(r)∗μB(v)≥min{μA(r),μB(v)}≠0[Since,μA(r)≠0andμB(v)≠0]. |
and
γA⊗B(a)=∗a=st(γA(s)⋄γB(t))≤γA(r)⋄γB(nv)≤γA(r)⋄γB(v)≤max{γA(r),γB(v)}≠1[Since,γA(r)≠1andγB(v)≠1]. |
This is a contradiction with μλP(a)=0 and γλP(a)=1. Thus for any intuitionistic fuzzy normed ideals A and B of NR we have A∘B⊆λP imply that A⊆λP or B⊆λP. So, λP=(μλP,γλP) is an intuitionistic fuzzy normed prime ideal of NR.
Conversely, suppose λP is an intuitionistic fuzzy normed prime ideal. Let A and B be two intuitionistic fuzzy normed prime ideal of NR such that A∘B⊆P. Let r∈NR, suppose μλA⊛λB(r)≠0 and γλA⊗λB(r)≠1, then μλA⊛λB(r)=⋄r=cd(μλA(c)∗μλB(d))≠0 and γλA⊗λB(r)=∗r=cd(γλA(c)⋄γλB(d))≠1. Then we have c,d∈NR such that r=cd, μλA(c)≠0, μλB(d)≠0 and γλA(c)≠1, γλB(d)≠1. Then, μλA(c)=1, μλB(d)=1 and γλA(c)=0, γλB(d)=0. Which implies c∈A and d∈B, therefore r=cd∈A∘B⊆P. Then, μλP(r)=1 and γλP(r)=0. Thus, for all r∈NR, μλA⊛λB(r)≤μλP(r) and γλA⊗λB(r)≥γλP(r). So, λA∘λB⊆λP. Since λP is an intuitionistic fuzzy normed prime ideal. Then either λA⊆λP or λB⊆λP. Therefore, either A⊆P or B⊆P. Hence P is a prime ideal in NR.
Definition 4.7. [15] Given a ring R and a proper ideal M of R, M is a maximal ideal of R if any of the following equivalent conditions hold:
i. There exists no other proper ideal J of R so that M⊊J.
ii. For any ideal J with M⊆J, either J=M or J=R.
Definition 4.8. An intuitionistic fuzzy normed ideal A of a normed ring NR is said to be an intuitionistic fuzzy normed maximal ideal if for any intuitionistic fuzzy normed ideal B of NR, A⊆B, implies that either B∗=A∗ or B=λNR. Intuitionistic fuzzy normed maximal left (right) ideal are correspondingly specified.
Proposition 4.9. Let A be an intuitionistic fuzzy normed maximal left (right) ideal of NR. Then, ∣ImμA∣ = ∣ImγA∣=2
Theorem 4.10. Let A be an intuitionistic fuzzy normed maximal left (right) ideal of a normed ring NR. Then A∗={r∈NR:μA(r)=μA(0) and γA(r)=γA(0)} is a maximal left (right) ideal of NR.
Proof. As A is not constant, A∗≠NR. Then using Proposition 4.9, A is two-valued. Let Im(μA)={1,α} and Im(γA)={0,β}, where 0≤α<1 and 0<β≤1. Assume M to be a left ideal of NR in away that A∗⊆M. Take B be an intuitionistic fuzzy subset of NR where if r∈M then μB(r)=1 and γB(r)=0 and if r∉M then μB(r)=c and γB(r)=d, where α<c<1 and 0<d<β. Then B is an intuitionistic fuzzy normed left ideal. Obviously A⊆B. As A is an intuitionistic fuzzy normed maximal left ideal of NR then A∗=B∗ or B=λNR. If A∗=B∗ then A∗=M given that B∗=M. If B=λNR subsequently M=NR. Therefore, A∗ is a maximal left ideal of NR.
Theorem 4.11. If A is an intuitionistic fuzzy normed maximal left (right) ideal of NR, then μA(0)=1 and γA(0)=0.
Proof. Suppose μA(0)≠1 and γA(0)≠0 and B to be an intuitionistic fuzzy subset of NR defined as B={r∈NR:μB(r)=h and γB(r)=k}, where μA(0)<h<1 and 0<k<γA(0). Then, B is an intuitionistic fuzzy normed ideal of NR. We can simply check that A⊂B, B≠λNR and B∗={r∈NR:μB(r)=μB(0) and γB(r)=γB(0)}=NR. Hence, A⊂B but A∗≠B∗ and B≠λNR which contradicts with the assumption that A is an intuitionistic fuzzy normed maximal ideal of NR. Therefore, μA(0)=1 and γA(0)=0.
Theorem 4.12. Let A be a intuitionistic fuzzy normed left (right) ideal of NR. If A∗ is a maximal left (right) ideal of NR with μA(0)=1 and γA(0)=0, then A is an intuitionistic fuzzy normed maximal left (right) ideal of NR.
Proof. By Proposition 4.9 A is two-valued. Let Im(μA)={1,α} and Im(γA)={0,β}, where 0≤α<1 and 0<β≤1. Define B to be an intuitionistic fuzzy normed left ideal of NR where A⊆B. Hence, μB(0)=1 and γB(0)=0. Let r∈A∗. Then 1=μA(0)=μA(r)≤μB(r) and 0=γA(0)=γA(r)≥γB(r). Thus μB(r)=1=μB(0) and γB(r)=0=γB(0), hence r∈B∗ then A∗⊆B∗. Given that A∗ a maximal left ideal of NR, then A∗=B∗ or B∗=NR. If B∗=NR subsequently B=λNR. Therefore, A is an intuitionistic fuzzy normed maximal left ideal of NR.
Remark 4.13. Let A⊆NR and let 0≤α≤1 and 0≤β≤1. Let λAα,β be an intuitionistic fuzzy subset of NR where μλAα(r)=1 if r∈A, μλAα(r)=α if r∉A and γλAβ(r)=0 if r∈A, γλAβ(r)=β if r∉A. If α=0 and β=1, the λAα,β is the intuitionistic characteristic function of A, which identified by λA=(μλA,γλA). If NR is a ring and A is an intuitionistic fuzzy normed left (right) ideal of NR, then:
- μλAα(0)=1, γλAβ(0)=0;
- (λAα,β)∗=A, [(λAα,β)∗={r∈NR:μλAα(r)=μλAα(0), γλAβ(r)=γλAβ(0)}=A];
- Im(μA)={1,α} and Im(γA)={0,β};
- λAα,β is an intuitionistic fuzzy normed left (right) ideal of NR.
In this article, we defined the intrinsic product of two intuitionistic fuzzy normed ideals and proved that this product is a subset of their intersection. Also, we characterized some properties of intuitionistic fuzzy normed ideals. We initiated the concepts of intuitionistic fuzzy normed prime ideal and intuitionistic fuzzy normed maximal ideal and we established several results related to these ideals. Further, we specified the conditions under which a given intuitionistic fuzzy normed ideal is considered to be an intuitionistic fuzzy normed prime (maximal) ideal. We generalised the relation between the intuitionistic characteristic function and prime (maximal) ideals.
The author declares no conflict of interest in this paper
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