Topological Properties of Jordan Intuitionistic Fuzzy Normed Spaces

Vakeel A. Khan; Umme Tuba

doi:10.1515/ms-2023-0033

Article Open Access

Topological Properties of Jordan Intuitionistic Fuzzy Normed Spaces

Vakeel A. Khan and Umme Tuba

Published/Copyright: March 31, 2023

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From the journal Mathematica Slovaca Volume 73 Issue 2

Abstract

The article delves into the task of formulation of meticulous sequence spaces whose elements’ convergence is a generalized version of the Cauchy convergence in the setting of intuitionistic fuzzy norm. The task of attaining a finite limit is attained via an infinite matrix operator, namely, Jordan totient operator. We discuss some topological and algebraic properties and establish some inclusion relations between the spaces.

2020 Mathematics Subject Classification: Primary 40A05; 40A30; 40A35; 40C05; 03B52; 46A45; Secondary 40H05; 46A35

Keywords: Jordan quotient operator; t-norm; t-conorm; Intuitionistic fuzzy norm; Jordan 𝓘-convergence; compactness; completeness

1. Introduction

Researchers have been taking initiatives in proposing novel theories that suggests the convergence of sequences by either operating a function or transforming it into an entirely different sequence using a linear operator. This notion of redefining the sequences’ convergence which fail to converge in the usual sense falls under the category of summability theory. It has proved to be quite an indispensable tool to analyse the depths of analysis. One can perceive it as a map with domain and codomain as set of sequences and a finite limit, respectively. The introduction of regular summability method [38] ensures the convergence of the originally convergent sequence to its Cauchy limit. Several authors have deployed the notion of one such naturally occuring regular summability method; regular infinite matrix, see [12, 20, 28, 39]. For a given infinite matrix 𝒫 = (p_ab) and sequence spaces 𝒮, 𝒯, the map 𝒫: 𝒮 → 𝒯 defined by 𝒫(u) = ∑ p_abu_b for all u = (u_b) ∈ 𝒮 and a, b ∈ ℕ, is the linear transformation that acts on a sequence space to produce some finite limit. Readers inquisitive about articles on sequences spaces and summability theory by means of matrix transformation are advised to explore [11, 16, 20, 26, 27].

Thereafter the introduction of fuzzy norm as a generalization of usual norm by Katsaras [19] and Felbin [9]; Bag and Samanta [2] established the base of an even general fuzzy norm. The underlying idea was to define a function that interprets the degree of truth of the length of a vector instead of assuming the length to be a fixed specified scalar. Later it was Saadati and Vaezpour [32] who improvised on the conditions of fuzzy norm. In the wake of coming up with the definition of fuzzy norm defined in terms of degree of truth via membership function, Atanassov [1] pointed out a shortcoming in the fuzzy norm and proposed the set be defined using an additional function named non-membership function which defines the degree of falsity or non-belongingess of an element in a given set which he named as intuitionistic fuzzy set.

This introduction of a newfangled idea gave the researchers an insight into defining intuitionistic fuzzy metric space [30] and intuitionistic fuzzy normed space [31] which ultimately led to inception of sequences’ convergence, statistical convergence [18], lacunary statistical convergence [29], generalized ideal convergence [7] and summability method [33] in the underlying space.

One such definite matrix operator namely Jordan totient matrix operator denoted by ϒ^r was introduced in [15] via Jordan totient J_r function whose domain and codomain are ℕ. The function is the number of r tuples (h₁, h₂, …, h_r) such that 1 ≤ h_i ≤ n and gcd (h₁, h₂, …, h_r, n) = 1. It is defined as Jr(n)=nr∏p∣n(1−1pr) where h=p1α1,p2α2,…,pkαk for α ≥ 1 is the prime decomposition of n. Thereupon the Jordan totient matrix operator denoted by Υr=(νnkr) is defined as:

1.1 νnkr={Jr(k)nrif k|n,0otherwise,

and its inverse (ϒ^r)⁻¹ is given by

1.2 (Υr)−1={μ(nk)Jr(n)krif k|n,0otherwise,

where μ is the Möbius function defined as:

μ(n)={0 if p2|n for some prime p,1 if n=1,(−1)i if n=∏k=1ipk where pks are distinct.

Later Kara [17], İlkhan [14] and Khan [21] used the operator to study sequence spaces and expounded certain riveting results. In the ongoing article we define sequence spaces with the help of Jordan totient matrix operator in the setting of intuitionistic fuzzy normed spaces, study the ideal convergence of these sequences, present compelling counter examples and study algebraic and topological properties of these spaces.

2. Preliminaries

For two sequence spaces S,T and an infinite matrix 𝒫 = (p_nk), the 𝒫 transform of u = (u_k) is given by Pu={Pn(u)}n=1∞∈T, where

Pn(u)=∑k=1pnkuk,n∈N.

Lemma 2.1 ([38]).

An infinite matrix 𝒫 = (p_nk) is regular iff:

for every n ∈ ℕ, there exists ℳ > 0, such that ∑k|pnk|≤M,
limn→∞pnk=0 for all k ∈ ℕ,
limn→∞∑kpnk=1.

The article will employ the use of the Jordan totient infinite matrix operator Υr=(νnkr) which is defined as:

2.1 νnkr={Jr(k)nrif k|n,0otherwise.

Equivalently,

Υr=[Jr(1)100000…Jr(2)2rJr(1)2r0000…Jr(1)3r0Jr(3)3r000…Jr(1)4rJr(2)4r0Jr(4)4r00…Jr(1)5r000Jr(5)5r0…Jr(1)6rJr(2)6rJr(3)6r00Jr(6)6r…⋮⋮⋮⋮⋮⋮⋱]

The ϒ^r transform of (u_k) ∈ ω is defined as Υnr(u):=1nr∑Jr(k)uk.

Definition 1 ([25]).

A cluster of subsets ℐ of a non-empty set 𝒳 is said to be an ideal in 𝒳 if:

∅ ∈ ℐ,
A,B∈I⇒A∪B∈I,
A∈I,B⊆A⇒B∈I.

An ideal ℐ ⊆ 2^𝒳 such that ℐ ≠ 2^𝒳 is a nontrivial ideal. It transforms into an admissible ideal if ℐ contains each singleton subset of 𝒳 and into a maximal ideal if there does not exist any nontrivial ideal 𝒥 ≠ ℐ such that ℐ ⊂ 𝒥.

Definition 2 ([25]).

A cluster of subsets ℱ of a non-empty set 𝒳 is said to be a filter in 𝒳 if:

∅ ∉ ℱ,
A,B∈F⇒A∩B∈F,
𝒜 ∈ ℱ and ℬ ⊃ 𝒜 imply ℬ ∈ ℱ.

F(I)={K⊂X:Kc∈I} is the filter associated with ideal ℐ. All through the paper, we take ℐ as an admissible ideal in ℕ.

Definition 3 ([25]).

A sequence u = (u_k) ∈ ω is said to be ℐ-convergent to c ∈ ℂ if for every ε > 0,

2.2 {k∈N:|uk−c|≥ε}∈I.

The commonly used mathematical notation to depict the above definition of convergence is ℐ-lim u_k = c.

Definition 4 ([37]).

Let ℐ ⊆ P(ℕ) is a non trivial ideal, for any two sequences u_k and (v_k), we say u_k = v_k for a.a.k.r.ℐ if {k ∈ ℕ: u_k ≠ v_k} ∈ ℐ.

Definition 5 ([25]).

A sequence u = (u_k) ∈ ω is said to be ℐ-Cauchy if, for each ε > 0, there exists a number 𝒥 = 𝒥(ε) such that the set

{k∈N:|uk−uJ|≥ε}∈I.

Definition 6 ([34]).

A binary operation *: [0, 1] × [0, 1] → [0, 1] is said to be a continuous t-norm if:

* is associative and commutative,
* is continuous,
a * 1 = a for all a ∈ [0, 1],
a * b ≤ c * d whenever a ≤ c and b ≤ d for each a, b, c, d ∈ [0, 1].

Definition 7 ([34]).

A binary operation ⋄: [0, 1] × [0, 1] → [0, 1] is said to be a continuous t-conorm if:

⋄ is associative and commutative,
⋄ is continuous,
a ⋄ 0 = a for all a ∈ [0, 1],
a ⋄ b ≤ c ⋄ d whenever a ≤ c and b ≤ d for each a, b, c, d ∈ [0, 1].

Definition 8 ([31]).

Let 𝒳 is a linear space, * & ⋄ are continuous t-norm and t-conorm respectively and ϕ, ψ are fuzzy sets on 𝒳 × (0, ∞). The five-tuple (𝒳, ϕ, ψ, *, ⋄) is an intuitionistic fuzzy normed space if for every u, v ∈ 𝒳 and s, t > 0 the following conditions hold:

ϕ(u, t) + ψ(u, t) ≤ 1,
ϕ(u, t) > 0,
ϕ(u, t) = 1 if and only if u = 0,
ϕ(αu,t)=ϕ(u,tα) for each α ≠ 0,
ϕ(u, t) * ϕ(v, s) ≤ ϕ(u + v, t + s),
ϕ(u, ·): (0, ∞) → [0, 1] is continuous,
limt→∞ϕ(u,t)=1 and limt→0ϕ(u,t)=0,
ψ(u, t) < 1,
ψ(u, t) = 0 if and only if u = 0,
ψ(αu,t)=ψ(u,tα) for each α ≠ 0,
ψ(u, t) * ψ(v, s) ≥ ψ(u + v, t + s),
ψ(u, ·): (0, ∞) → [0, 1] is continuous,
limt→∞ψ(u,t)=0 and limt→0ψ(u,t)=1.

Consequently the pair (ϕ, ψ) is called intuitionistic fuzzy norm.

Definition 9 ([22]).

Let (𝒳, ϕ, ψ, *, ⋄) be an IFNS. A sequence u = (u_k) is said to be ℐ-convergent to l ∈ 𝒳 with respect to intuitionistic fuzzy norms (ϕ, ψ), if for every ε > 0 and t > 0, the set

{k∈N:ϕ(uk−l,t)≤1−ε or ψ(uk−l,t)≥ε}∈I.

The notation ℐ^{(ϕ, ψ)}-lim u_k = l will be used in the article to denote the ideal convergence of the sequence (u_k) to l with respect to the intuitionistic fuzzy norm (φ, ψ).

Definition 10 ([29]).

Let (𝒳, ϕ, ψ, *, ⋄) be an IFNS. A sequence u = (u_k) is said to be ℐ-Cauchy sequence with respect to (ϕ, ψ), if for every ε > 0 and t > 0, there exists 𝒩 = 𝒩(ε) ∈ ℕ such that the set

{k∈N:ϕ(uk−uN,t)≤1−ε or ψ(uk−uN,t)≥ε}∈I.

Definition 11.

Let (𝒳, ϕ, ψ, *, ⋄) be an IFNS. Then (𝒳, ϕ, ψ, *, ⋄) is said to be complete if every Cauchy sequence is convergent with respect to the intuitionistic fuzzy norm (ϕ, ψ).

3. Main results

Through the whole of the current section we presume the ideal ℐ to be a nontrivial admissible ideal of subset of ℕ. We set out the following sequence spaces:

3.1 c(ϕ,ψ)I(Υr)={u=(uk)∈ω:{n∈N:for some l∈C,ϕ(Υnr(u)−l,t)≤1−ε or ψ(Υnr(u)−l,t)≥ε}∈I},

3.2 c0(ϕ,ψ)I(Υr)={u=(uk)∈ω:{n∈N: ϕ(Υnr(u),t)≤1−ε or ψ(Υnr(u),t)≥ε}∈I},

3.3 ℓ∞(ϕ,ψ)I(Υr)={u=(uk)∈ω:{n∈N: ∃ ξ∈(0,1),ϕ(Υnr(u),t)≤1−ξ or ψ(Υnr(u),t)≥ξ}∈I}.

3.4 ℓ∞(ϕ,ψ)(Υr)={u=(uk)∈ω:{n∈N: ∃ ξ∈(0,1), ϕ(Υnr(u),t)≥1−ξ or ψ(Υnr(u),t)≤ξ}.

We bring forth the following definitions of open ball and closed ball with centre at u and radius r > 0 with respect to parameter of fuzziness ε ∈ (0, 1) as follows:

3.5 BuI(r,ε)(Υr)={v=(vk)∈ω:{n∈N:ϕ(Υnr(u)−Υnr(v),r)≤1−ε or ψ(Υnr(u)−Υnr(v),r)≥ε}∈I}

3.6 BuI[r,ε](Υr)={v=(vk)∈ω:{n∈N:ϕ(Υnr(u)−Υnr(v),r)<1−ε or ψ(Υnr(u)−Υnr(v),r)>ε}∈I}.

Theorem 3.1.

The spaces c0(ϕ,ψ)I(Υr) and c(ϕ,ψ)I(Υr) are linear spaces.

Proof.

The proof of linearity of the space c0(ϕ,ψ)I(Υr) can be conclusively drawn on parallels of c(ϕ,ψ)I(Υr). Given arbitrary sequences a=(ak),b=(bk)∈c(ϕ,ψ)I(Υr) imply existence of a₀, b₀ ∈ ℂ so that (a_k) and (b_k) ℐ–converge to a₀ and b₀, respectively. For t > 0, 0 < ε < 1 and γ, λ ∈ ℝ, consider the subsequent sets:

A={n∈N:ϕ(Υnr(a)−a0,t2|γ|)≤1−ε or ψ(Υnr(a)−a0,t2|γ|)≥ε}∈I,Ac={n∈N:ϕ(Υnr(a)−a0,t2|γ|)>1−ε or ψ(Υnr(a)−a0,t2|γ|)<ε}∈F(I),B={n∈N:ϕ(Υnr(b)−b0,t2|λ|)≤1−ε or ψ(Υnr(b)−b0,t2|λ|)≥ε}∈I,Bc={n∈N:ϕ(Υnr(b)−b0,t2|λ|)>1−ε or ψ(Υnr(a)−a0,t2|λ|)<ε}∈F(I).

The set 𝒞 = 𝒜^c ∩ ℬ^c being a non-empty set lies in ℱ(ℐ), so consider n ∈ 𝒞, then

ϕ(Υnr(γa+λb)−(γa0+λb0),t)≥ϕ(γΥnr(a)−γa0,t2)∗ϕ(λΥnr(b)−λb0,t2)=ϕ(Υnr(a)−a0,t2|γ|)∗ϕ(Υnr(b)−b0,t2|λ|)>(1−ε)∗(1−ε)=1−ε

⟹ϕ(Υnr(γa+λb)−(γa0+λb0),t)>1−ε

and

ψ(Υnr(γa+λb)−(γa0+λb0),t)≤ψ(γΥnr(a)−γa0,t2)⋄ψ(λΥnr(b)−λb0,t2)=ψ(Υnr(a)−a0,t2|γ|)⋄ψ(Υnr(b)−b0,t2|λ|)<ε⋄ε=ε

⟹ψ(Υnr(γa+λb)−(γa0+λb0),t)<ε.

Thereupon, we conclude

C⊂ {n∈N:ϕ(Υnr(γa+λb)−(γa0+λb0),t)>1−ε}or {n∈N:ψ(Υnr(γa+λb)−(γa0+λb0),t)<ε}.

By employing the properties of ℱ(ℐ), we have

{n∈N:ϕ(Υnr(γa+λb)−(γa0+λb0),t)>1−ε or ψ(Υnr(γa+λb)−(γa0+λb0),t)<ε}∈F(I),

which implies that the sequence (γa_k + λb_k) ℐ-converges to γa₀ + λb₀. Therefore, (γak+λbk)∈c(ϕ,ψ)I(Υr). Hence, c(ϕ,ψ)I(Υr) is a linear space. □

Theorem 3.2.

The inclusion relation c0(ϕ,ψ)I(Υr)⊂c(ϕ,ψ)I(Υr)⊂c∞(ϕ,ψ)I(Υr) holds.

Proof.

The inclusion of c0(ϕ,ψ)I(Υr) in c(ϕ,ψ)I(Υr) is pretty evident. We demonstrate that c(ϕ,ψ)I(Υr)⊂c∞(ϕ,ψ)I(Υr). Consider the sequence u=(uk)∈c(ϕ,ψ)I(Υr). Then there exists l ∈ ℂ such that ℐ_{(ϕ, ψ)}(ϒ^r)-lim(u_k) = l, and for every 0 < ε < 1 and t > 0, the set

X={n∈N:ϕ(Υnr(u)−l,t2)>1−ε or ψ(Υnr(u)−l,t2)<ε}∈F(I).

Let ϕ(l,t2)=p and ψ(l,t2)=q where p, q ∈ (0, 1), t > 0 and 0 < ε < 1, there exist c, d ∈ (0, 1) such that (1 − ε) * p > 1 − c and ε ⋄ q < d. So for n ∈ 𝒳, we have

ϕ(Υnr(u),t)=ϕ(Υnr(u)−l+l,t)≥ϕ(Υnr(u)−l,t2)∗ϕ(l,t2)>(1−ε)∗p>1−c

and

ψ(Υnr(u),t)=ψ(Υnr(u)−l+l,t)≤ψ(Υnr(u)−l,t2)⋄ψ(l,t2)<(1−ε)⋄q<d.

Taking h = max{c, d}, we have

{n∈N,∃ h∈(0,1):ϕ(Υnr(u)−l,t)>1−h or ψ(Υnr(u)−l,t)<h}∈F(I)

⟹u=(uk)∈c∞(ϕ,ψ)I(Υr).

□

The converse of the inclusion relation does not hold. We present the following examples in support of our claim.

Example 1.

Let (ℝ, ||.||) be a normed space equipped with supremum norm, a * b = min{a, b} and a ⋄ b = max{a, b} for all a, b ∈ (0, 1). Consider the norms (ϕ, ψ) on 𝒳² × (0, ∞) as follows:

ϕ(u,t)=tt+‖u‖ and ψ(u,t)=‖u‖t+‖u‖.

Then (ℝ, ϕ, ψ, *, ⋄) is a standard IFNS. Consider the sequence (uk)={u0+1k} where u₀ ≠ 0 ∈ ℝ. The sequence (u_k) distinctly lies in c(ϕ,ψ)I(Υr)∖c0(ϕ,ψ)I(Υr).

Example 2.

Let (ℝ, ||·||) be the normed space equipped with the intutionistic fuzzy norms (ϕ, ψ) as aforementioned above. Consider the sequence (uk)=sin⁡(1k). Then, (uk)∈c∞(ϕ,ψ)I(Υr)∖c(ϕ,ψ)I(Υr).

Theorem 3.3.

Every open ball with centre at z and radius r > 0 with respect to parameter of fuzziness 0 < ε < 1 i.e., BzI(r,ε)(Υr) is an open set in c(ϕ,ψ)I(Υr).

Proof.

Consider the open ball with centre at z and radius r > 0 with parameter of fuzziness 0 < ε < 1,

BzI(r,ε)(Υnr)={y=(yk)∈ω:{n∈N:ϕ(Υnr(z)−Υnr(y),r)≤1−ε or ψ(Υnr(z)−Υnr(y),r)≥ε}∈I}⟹BzI(r,ε)(Υnr)={y=(yk)∈ω:{n∈N:ϕ(Υnr(z)−Υnr(y),r)>1−ε or ψ(Υnr(z)−Υnr(y),r)<ε}∈F(I)}.

Consider the element y=(yk)∈BzI(r,ε)(Υnr). Then its corresponding set

{n∈N:ϕ(Υnr(z)−Υnr(y),r)>1−ε or ψ(Υnr(z)−Υnr(y),r)<ε}∈F(I).

For ϕ(Υnr(z)−Υnr(y),r)>1−ε and ψ(Υnr(z)−Υnr(y),r)<ε, there exists r₀ ∈ (0, r) such that ϕ(Υnr(z)−Υnr(y),r0)>1−ε and ψ(Υnr(z)−Υnr(y),r0)<ε. Setting ε0=ϕ(Υnr(z)−Υnr(y),r0) we get ε₀ > 1 − ε which further concludes the existence of an element s ∈ (0, 1) such that ε₀ > 1 − s > 1 − ε. For a given ε₀ > 1 − s, we can find ε₁, ε₂ ∈ (0, 1) such that ε₀ * ε₁ > 1 − s and (1 − ε₀) ⋄ (1 − ε₂) < s. Assume ε₃ = max{ε₁, ε₂}. Consider the open ball ByI(r−r0,1−ε3)(Υnr). The containment of ByI(r−r0,1−ε3)(Υnr) in BzI(r,ε)(Υnr) will give us the desired result.

Let w=(wk)∈ByI(r−r0,1−ε3)(Υnr), then {n∈N:ϕ(Υnr(y)−Υnr(w),r−r0)>ε3 or ψ(Υnr(y)−Υnr(w),r−r0)<1−ε3}∈F(I). Therefore,

ϕ(Υnr(z)−Υnr(w),r)≥ϕ(Υnr(z)−Υnr(y),r0)∗ϕ(Υnr(y)−Υnr(w),r−r0)≥ε0∗ε3≥ε0∗ε1>(1−s)>(1−ε)

⟹{n∈N:ϕ(Υnr(z)−Υnr(w),r)>1−ε}∈F(I),

and correspondingly

ψ(Υnr(z)−Υnr(w),r)≤ψ(Υnr(z)−Υnr(y),r0)⋄ψ(Υnr(y)−Υnr(w),r−r0)≤(1−ε0)⋄(1−ε3)≤(1−ε0)⋄(1−ε2)<s<ε

⟹{n∈N:ψ(Υnr(z)−Υnr(w),r)<ε}∈F(I).

Thus the set

{n∈N:ϕ(Υnr(z)−Υnr(w),r)>1−ε or ψ(Υnr(z)−Υnr(w),r)<ε}∈F(I)

⟹ByI(r−r0,1−ε3)(Υnr)⊂BzI(r,ε)(Υnr).

□

Remark 1.

The spaces c0(ϕ,ψ)I(Υr) and c(ϕ,ψ)I(Υr) are IFNS with respect to intuitionistic fuzzy norms (ϕ, ψ).

Remark 2.

τ(ϕ,ψ)I(Υr)={A⊂c(ϕ,ψ)I(Υr):for each z=(zk)∈A, there exist r>0 and ε∈(0,1) such that BzI(r,ε)(Υnr)⊂A}. Then τ(ϕ,ψ)I(Υr) defines a topology on the sequence space τ(ϕ,ψ)I(Υr). The collection defined by B={BzI(r,ε):z∈c(ϕ,ψ)I(Υr),r>0 and ε∈(0,1)} is a base for the topology τ(ϕ,ψ)I(Υr) on the space c(ϕ,ψ)I(Υr).

Theorem 3.4.

The spaces c0(ϕ,ψ)I(Υr) and c(ϕ,ψ)I(Υr) are Hausdorff spaces.

Proof.

Let v = (v_k) and w=(wk)∈c0(ϕ,ψ)I(Υr) such that v ≠ w. Then for each n ∈ ℕ and r > 0, implies 0<ϕ(Υnr(v)−Υnr(w),r)<1 and 0<ψ(Υnr(v)−Υnr(w),r)<1.

Putting ε1=ϕ(Υnr(v)−Υnr(w),r), ε2=ψ(Υnr(v)−Υnr(w),r) and ε = max{ε₁, 1 − ε₂}. Then for each ε₀ > ε, there exist ε₃, ε₄ ∈ (0, 1) such that ε₃ * ε₃ ≥ ε₀ and (1 − ε₄) ⋄ (1 − ε₄) ≤ (1 − ε₀). Assigning ε₅ = max{ε₃, ε₄}, consider the open balls BvI(1−ε5,r2)(Υnr) and BwI(1−ε5,r2)(Υnr) centered at v and w respectively. We demonstrate that BvI(1−ε5,r2)(Υnr)∩BwI(1−ε5,r2)(Υnr)=ϕ. If viable let z=(zk)∈BvI(1−ε5,r2)(Υnr)∩BwI(1−ε5,r2)(Υnr). Then for the set {n ∈ ℕ} ∈ ℱ(ℐ), we have

3.7 ε1=ϕ(Υnr(v)−Υnr(w),r)≥ϕ(Υnr(v)−Υnr(z),r2)∗ϕ(Υnr(z)−Υnr(w),r2)>ε5∗ε5≥ε3∗ε3≥ε0>ε1

and

3.8 ε2=ψ(Υnr(v)−Υnr(w),r)≤ψ(Υnr(v)−Υnr(z),r2)⋄ψ(Υnr(z)−Υnr(w),r2)<(1−ε5)⋄(1−ε5)≤(1−ε4)⋄(1−ε4)≤(1−ε0)<ε2.

Equation (3.7) leads to contradiction. Therefore, BvI(1−ε5,r2)(Υnr)∩BwI(1−ε5,r2)(Υnr)=ϕ. Hence, the space c(ϕ,ψ)I(Υr) is a Hausdorff space. □

Theorem 3.5.

If a sequence z = (z_k) ∈ ω is Jordan intuitionistic fuzzy ℐ convergent then the I(ϕ,ψ)(Υnr)-limit is unique.

Proof.

Presuming the sequence z = (z_k) to be Jordan intuitionistic fuzzy ℐ convergent with non-identical ideal limits l₁ and l₂. Given an ε ∈ (0, 1), there exists ε₁ ∈ (0, 1) such that (1 − ε₁) * (1 − ε₁) > 1− ε and ε₁ ⋄ ε₁ < ε. Thus the sets

S={n∈N:ϕ(Υnr(z)−l1,t2)≤1−ε1 or ψ(Υnr(z)−l1,t2)≥ε1}∈I⟹Sc={n∈N:ϕ(Υnr(z)−l1,t2)>1−ε1 or ψ(Υnr(z)−l1,t2)<ε1}∈F(I),T={n∈N:ϕ(Υnr(z)−l2,t2)≤1−ε1 or ψ(Υnr(z)−l2,t2)≥ε1}∈I⟹Tc={n∈N:ϕ(Υnr(z)−l2,t2)>1−ε1 or ψ(Υnr(z)−l2,t2)<ε1}∈F(I).

Then 𝒮^c ∩ 𝒯^c ≠ ϕ. Taking n ∈ 𝒮^c ∩ 𝒯^c, we have

ϕ(l1−l2,t)≥ϕ(Υnr(z)−l1,t2)∗ϕ(Υnr(z)−l2,t2)>(1−ε1)∗(1−ε1)>(1−ε)

and

ψ(l1−l2,t)≤ψ(Υnr(z)−l1,t2)⋄ψ(Υnr(z)−l2,t2)<ε1⋄ε1<ε.

ε ∈ (0, 1) being arbitrary; l₁ = l₂. That is I(ϕ,ψ)(Υnr)-limit is unique. □

Theorem 3.6.

A sequence z = (z_k) ∈ ω is Jordan intuitionistic fuzzy ℐ convergent with respect to intuitionistic fuzzy norms (ϕ, ψ) iff it is Jordan intuitionistic fuzzy ℐ-Cauchy with reference to the identical norms.

Proof.

Let z = (z_k) ∈ ω is Jordan intuitionistic fuzzy ℐ convergent with respect to intuitionistic fuzzy norms (ϕ, ψ) such that I(ϕ,ψ)(Υnr)-lim(z_k) = l and there exists ε₁ ∈ (0, 1) such that (1 – ε₁) * (1 – ε₁) > 1 – ε and ε₁ ⋄ ε₁ < ε for a given ε ∈ (0, 1). Thus for all t > 0,

P={n∈N:ϕ(Υnr(z)−l1,t)≤1−ε1 or ψ(Υnr(z)−l1,t)≥ε1}∈I⟹Pc={n∈N:ϕ(Υnr(z)−l1,t)>1−ε1 or ψ(Υnr(z)−l1,t)<ε1}∈F(I).

For n ∈ 𝒫^c, ϕ(Υnr(z)−l1,t)>1−ε1 or ψ(Υnr(z)−l1,t)<ε1. For a particular k ∈ 𝒫^c, we can say

Q={n∈N:ϕ(Υnr(z)−Υkr(z),t)≤1−ε or ψ(Υnr(z)−Υkr(z),t)≥ε}.

Let

n∈Q ⟹ ϕ(Υnr(z)−Υkr(z),t)≤1−ε or ψ(Υnr(z)−Υkr(z),t)≥ε.

On the contrary, let ϕ(Υnr(z)−Υkr(z),t)>1−ε. Then

1−ε≥ϕ(Υnr(z)−Υkr(z),t)≥ϕ(Υnr(z)−l,t2)∗ϕ(Υkr(z)−l,t2)>(1−ε1)∗(1−ε1)>(1−ε),

which is a contradiction. Likewise, consider ψ(Υnr(z)−Υkr(z),t)≥ε such that ψ(Υnr(z)−l,t2)≥ε1. On the contrary, let ψ(Υnr(z)−l,t2)<ε1. Hence

ε≤ψ(Υnr(z)−Υkr(z),t)≤ψ(Υnr(z)−l,t2)⋄ψ(Υkr(z)−l,t2)<ε1⋄ε1<ε,

which is a contradiction too. Therefore, for n ∈ 𝒬, we have ϕ(Υnr(z)−l,t)≤1−ε1 and ϕ(Υnr(z)−l,t)≥ε1, which imply n ∈ 𝒫. Therefore, 𝒬 ⊂ 𝒫 and 𝒬 ∈ ℐ. Consequently the sequence z = (z_k) is Jordan intuitionistic fuzzy ℐ-Cauchy with respect to norms (ϕ, ψ).

Conversely, presuppose z = (z_k) is Jordan intuitionistic fuzzy ℐ- Cauchy with respect to the norms (ϕ, ψ) and is not Jordan intuitionistic fuzzy ℐ-convergent. As a result, there exists k ∈ ℕ such that

A={n∈N:ϕ(Υnr(z)−Υkr(z),t)≤1−ε or ψ(Υnr(z)−Υkr(z),t)≥ε}∈I

and

B={n∈N:ϕ(Υnr(z)−l,t)>1−ε or ψ(Υnr(z)−l,t)<ε}∈I

⟹1−ε≥ϕ(Υnr(z)−Υkr(z),t)≥ϕ(Υnr(z)−l,t2)∗ϕ(Υnr(z)−l,t2)>(1−ε1)∗(1−ε1)>1−ε.

Simultaneously,

ε≤ψ(Υnr(z)−Υkr(z),t)≤ψ(Υnr(z)−l,t2)⋄ψ(Υnr(z)−l,t2)<ε1⋄ε1<ε,

which lead to contradiction. Therefore, ℬ ∈ ℱ(ℐ) and hence, z = (z_k) is Jordan intuitionistic fuzzy ℐ-convergent. □

Theorem 3.7.

Consider IFNS c(ϕ,ψ)I(Υr) and τ(ϕ,ψ)I(Υr) be the topology on c(ϕ,ψ)I(Υr). Let (zj)=(zkj)j=1∞ be a sequence in c(ϕ,ψ)I(Υr). The sequence z^j → z as j → ∞ if and only if ϕ(Υnr(zj)−Υnr(z),t)→1 and ψ(Υnr(zj)−Υnr(z),t)→0 as n → ∞.

Proof.

Let z^j → z as j → ∞. Fix a designated r > 0 and 0 < ε < 1, there exists n ∈ ℕ such that (zj)∈BzI(r,ε)(Υnr) for all j ≥ k. Then,

S={n∈N:ϕ(Υnr(zj)−Υnr(z),r)≤1−ε or ψ(Υnr(zj)−Υnr(z),r)≥ε}∈I,

or equivalently,

Sc={n∈N:ϕ(Υnr(zj)−Υnr(z),r)>1−ε or ψ(Υnr(zj)−Υnr(z),r)<ε}∈F(I).

For {n∈N}⊆Sc,ϕ(Υnr(zj)−Υnr(z),r)>1−ε ⟹1−ϕ(Υnr(zj)−Υnr(z),r)<ε and ψ(Υnr(zj)−Υnr(z),r)<ε. Therefore, for n → ∞, 1−ϕ(Υnr(zj)−Υnr(z),r)→0 and ψ(Υnr(zj)−Υnr(z),r)→0. This implies that ϕ(Υnr(zj)−Υnr(z),r)→1 and ψ(Υnr(zj)−Υnr(z),r)→0 as n → ∞.

Conversely, suppose that for each t > 0, ϕ(Υnr(zj)−Υnr(z),t)→1 and ψ(Υnr(zj)−Υnr(z),t)→0 as n → ∞. Then for each ε ∈ (0, 1), there exists k ∈ ℕ such that 1−ϕ(Υnr(zj)−Υnr(z),t)<ε and ψ(Υnr(zj)−Υnr(z),t)<ε for all n≥k⟹ϕ(Υnr(zj)−Υnr(z),t)>1−ε and ψ(Υnr(zj)−Υnr(z),t)<ε for all n ≥ k. Consider the ideal ℐ generated by the set {n ∈ ℕ : n < k}, implies that the collection of sets generated by the set {n ∈ ℕ : n ≥ k} belongs to ℱ(ℐ). Thus {n∈N:ϕ(Υnr(zj)−Υnr(z),t)>1−ε or ϕ(Υnr(zj)−Υnr(z),t)<ε}∈F(I)⟹(zj)∈BzI(r,ε)(Υnr) for all n ≥ k. Hence, z^j → z as j → ∞. □

Theorem 3.8.

Let z=(zk)∈c(ϕ,ψ)I(Υr). Then for some l ∈ ℂ, zk→I(ϕ,ψ)(Υr)l if and only if for every ε ∈ (0, 1) and t > 0, there exist positive integers 𝒩 = 𝒩(z, ε, t) such that

{N∈N:ϕ(ΥNr(z)−l,t2)>1−ε or ψ(ΥNr(z)−l,t2)<ε}∈F(I).

Proof.

Suppose zk→I(ϕ,ψ)(Υr)l for some l ∈ ℂ. For given ε ∈ (0, 1), there exists r ∈ (0, 1) such that (1 – ε) * (1 – ε) > 1 – r and ε ⋄ ε < r. Since zk→I(ϕ,ψ)(Υr)l, for all t > 0,

X={n∈N:ϕ(Υnr(z)−l,t2)≤1−ε or ψ(Υnr(z)−l,t2)≥ε}∈I;

which implies that

Xc={n∈N:ϕ(Υnr(z)−l,t2)>1−ε or ψ(Υnr(z)−l,t2)<ε}∈F(I).

Conversely, let us choose 𝒩 ∈ 𝒳^c. Then

ϕ(ΥNr(z)−l,t2)>1−ε or ψ(ΥNr(z)−l,t2)<ε.

We show the existence of a positive integer 𝒩 = 𝒩(z, ε, t) such that

P={n∈N:ϕ(Υnr(z)−ΥNr(z),t)≤1−r or ψ(Υnr(z)−ΥNr(z),t)≥r}∈I.

We shall show that 𝒫 ⊆ 𝒳. On the contrary, let 𝒫 ⊈ 𝒳, i.e., there exists n ∈ 𝒫 such that n ∉ 𝒳. Then

ϕ(Υnr(z)−ΥNr(z),t)≤1−r and ϕ(Υnr(z)−l,t2)>1−ε.

Particularly,

ϕ(ΥNr(z)−l,t2)>1−ε.

Therefore, we have

1−r≥ϕ(Υnr(z)−ΥNr(z),t)≥ϕ(Υnr(z)−l,t2)∗ϕ(ΥNr(z)−l,t2)≥ (1−ε)∗(1−ε)>1−r

which is a contradiction. Similarly,

ψ(Υnr(z)−ΥNr(z),t)≥r and ψ(Υnr(z)−l,t)<ε.

Particularly,

ψ(ΥNr(z)−l,t2)<ε.

Therefore,

r≤ψ(Υnr(z)−ΥNr(z),t)≤ψ(Υnr(z)−l,t2)⋄ψ(ΥNr(z)−l,t2)≤ε⋄ε<r,

which is again a contradiction. Hence, 𝒫 ⊆ 𝒳 and since 𝒳 ∈ ℐ, implies 𝒫 ∈ ℐ. □

Definition 12.

A non-empty set S⊂c(ϕ,ψ)I(Υr) is compact if every open cover of 𝒮 defined by the open set of τ(ϕ,ψ)I(Υr) has a finite subcover.

Theorem 3.9.

Every finite subset 𝒮 of c(ϕ,ψ)I(Υr) is compact.

Proof.

Let 𝒮 = {z₁, z₂, z₃,…, z_n} be the finite subset of c(ϕ,ψ)I(Υr). For r > 0 and 0 < ε < 1, let us assume {BzI(r,ε)(Υnr):z∈S} is an open cover of 𝒮. Then S⊆⋃z∈SBzI(r,ε)(Υnr).

Now for all z_i ∈ 𝒮, i = 1, 2, 3,… n, we have zi∈⋃zi∈sBziI(r,ε)(Υnr). That implies zi∈BzjI(r,ε)(Υnr) for some j ∈ {1, 2, 3,…, n}. Then {BziI(r,ε)(Υnr):i=1,2,3,…,n} is a finite subcover of 𝒮. □

Theorem 3.10.

A set S⊆c(ϕ,ψ)I(Υr) is compact iff every sequence in 𝒮 has a convergent subsequence.

Proof.

Suppose 𝒮 is a compact subset of c(ϕ,ψ)I(Υr). Let (zkj)=(zj)j=1∞ be a sequence in 𝒮. For given 0 < ε < 1 and r > 0, let {BzI(r3,ε)(Υnr):z=(zk)∈S} be an open cover of 𝒮. This implies, (zj)∈⋃z∈S{BzI(r3,ε)(Υnr)}. Then there exists some z = (z_k) ∈ S such that (zj)∈BzI(r3,ε)(Υnr). Therefore, the set

X={n∈N:ϕ(Υnr(zj)−Υnr(z),fr3)>1−ε or ψ(Υnr(zj)−Υnr(z),r3)<ε}∈F(I).

There exists a finite subcover {BziI(r3,ε)(Υnr):zi∈S and i=1,2,3,….m} since 𝒮 is compact such that S⊆⋃i=1mBziI(r3,ε)(Υnr). Let (z^j_p) be a subsequence of (z^j). Then (zjp)∈⋃i=1mBziI(r3,ε)(Υnr), implies (zjp)∈BziI(r3,ε)(Υnr), for some z_i ∈ 𝒮. Therefore, the set

Y={n∈N:ϕ(Υnr(zjp)−Υnr(zi),r3)>1−ε or ψ(Υnr(zjp)−Υnr(zi),r3)<ε}∈F(I).

For n ∈ 𝒳 ∩ 𝒴,

ϕ(Υnr(zjp)−Υnr(z),r)≥ϕ(Υnr(zjp)−Υnr(zi),r3)∗ϕ(Υnr(zj)−Υnr(zi),r3)∗ϕ(Υnr(zj)−Υnr(z),r3)>(1−ε)∗(1−ε)∗(1−ε)=(1−ε).

Simultaneously,

ψ(Υnr(zjp)−Υnr(z),r)≤ψ(Υnr(zjp)−Υnr(zi),r3)⋄ψ(Υnr(zj)−Υnr(zi),r3)⋄ψ(Υnr(zj)−Υnr(z),r3)<ε⋄ε⋄ε=ε.

Take ε=1n. Then

limn→∞ϕ(Υnr(zjp)−Υnr(z),r)=limn→∞1−1n=1 and limn→∞ψ(Υnr(zjp)−Υnr(z),r)=limn→∞1n=0.

Thus by theorem (3.7), z^j_p → z, as p → ∞. Conversely, suppose (z^j_p) be the subsequence of a sequence (z^j) in 𝒮 such that (z^j_p) → z in 𝒮. Let on contrary 𝒮 is not a compact subset of c(ϕ,ψ)I(Υr). Let {BzI(r3,ε)(Υnr):x∈S} be an open cover of 𝒮 implies S⊆⋃z∈S{BzI(r3,ε)(Υnr). Therefore, the set

{n∈N:ϕ(Υnr(zj)−Υnr(z),r)>1−ε or ψ(Υnr(zj)−Υnr(z),r)<ε}∈F(I).

Since 𝒮 is not compact, there exists a finite subcover {BziI(r,ε)(Υnr):zi∈S,i=1,2,3…,m} such that S⊈⋃zi∈SBziI(r,ε)(Υnr), which implies that the set

{n∈N:ϕ(Υnr(zjp)−Υnr(zi),r)>1−ε or ψ(Υnr(zjp)−Υnr(zi),r)<ε}∉F(I)

⇒ for any 0 < ε < 1 and r > 0, (zjp)∉BzI(r,ε). Hence, (z^j_p) z, which is a contradiction. Hence, S is compact. □

Theorem 3.11.

Consider the IFNS c(ϕ,ψ)I(Υr). Let r > 0 and ε, ε′ ∈ (0, 1) such that (1 − ε) * (1 − ε) ≥ (1 − ε′) and ε ⋄ ε ≤ ε′. Then for any z=(zk)∈c(ϕ,ψ)I(Υr), BzI(r2,ε)(Υnr)¯⊆BzI(r,ε′)(Υnr).

Proof.

Let q=(qk)∈BzI(r2,ε)(Υnr)¯ and BqI(r2,ε)(Υnr) be an open ball with centre at q and radius ε. Hence, BzI(r2,ε)(Υnr)∩BqI(r2,ε)(Υnr)≠ϕ. Suppose v=(vk)∈BqI(r2,ε)(Υnr)∩BzI(r2,ε)(Υnr). Then, the sets

X={n∈N:ϕ(Υnr(q)−Υnr(v),r2)>1−ε1 or ψ(Υnr(q)−Υnr(v),r2)<ε1}∈F(I),Y={n∈N:ϕ(Υnr(z)−Υnr(v),r2)>1−ε or ψ(Υnr(z)−Υnr(v),r2)<ε1}∈F(I).

Consider n ∈ 𝒳 ∩ 𝒴. Then

ϕ(Υnr(z)−Υnr(q),r)≥ϕ(Υnr(z)−Υnr(v),r2)∗ϕ(Υnr(q)−Υnr(v),r2)>(1−ε)∗(1−ε)≥(1−ε′)

and

ψ(Υnr(z)−Υnr(q),r)≤ψ(Υnr(z)−Υnr(v),r2)⋄ψ(Υnr(q)−Υnr(v),r2)<ε⋄ε≤ε′.

Therefore, the set

{n∈N:ϕ(Υnr(z)−Υnr(q),r)>1−ε′ or ψ(Υnr(z)−Υnr(q),r)<ε′}∈F(I)

⟹q=(qk)∈BzI(r,ε′)(Υnr).

Hence, BzI(r2,ε)(Υnr)¯⊆BzI(r2,ε′)(Υnr). □

Theorem 3.12.

Let z = (z_k) ∈ ω. If there exists a sequence z′=(zk′)∈c(ϕ,ψ)I(Υr) such that Υnr(z)=Υnr(z′) for almost all n relative to ℐ, then z∈c(ϕ,ψ)I(Υr).

Proof.

Suppose Υnr(z)=Υnr(z′) for almost all n relative to ℐ. Then {n∈N:Υnr(z)≠Υnr(z′)}∈I. This implies {n∈N:Υnr(z)=Υnr(z′)}∈F(I). Therefore, for n ∈ ℱ(ℐ) for all t > 0,

ϕ(Υnr(z)−Υnr(z′),t2)=1 and ψ(Υnr(z)−Υnr(z′),t2)=0.

Since (zk′)∈c(ϕ,ψ)I(Υr), let I(ϕ,ψ)(Υr)−lim(zk′)=l. Then for every ε ∈ (0, 1) and t > 0,

X={n∈N:ϕ(Υnr(z′)−l,t2)>1−ε or ψ(Υnr(z′)−l,t2)<ε}∈F(I).

Consider the set

Y={n∈N:ϕ(Υnr(z)−l,t)>1−ε or ψ(Υnr(z)−l,t)<ε}.

We show that 𝒳 ⊂ 𝒴. So for n ∈ 𝒳, we have

ϕ(Υnr(z)−l,t)≥ϕ(Υnr(z)−Υnr(z′),t2)∗ϕ(Υnr(z′)−l,t2)>1∗(1−ε)=1−ε

and

ψ(Υnr(z)−l,t)≤ψ(Υnr(z)−Υnr(z′),t2)⋄ψ(Υnr(z′)−l,t2)<0⋄ε=ε.

This implies that n ∈ 𝒴 and hence, 𝒳 ⊂ 𝒴. Since 𝒳 ∈ ℱ(ℐ), therefore, 𝒴 ∈ ℱ(ℐ). Hence, z=(zk)∈c(ϕ,ψ)I(Υr). □

Theorem 3.13.

A closed ball BzI[r,ε](Υr) is a closed set in c(ϕ,ψ)I(Υr).

Proof.

Let y = (y_k) ∈ ω be such that y∈BzI[r,ε](Υr)¯. Thus there exists a sequence (yj)=(ykj)∈BzI[r,ε](Υr) such that y^j → y as j → ∞. Thus

X={n∈N:ϕ(Υnr(yj)−Υnr(z),r)≥1−ε or ψ(Υnr(yj)−Υnr(z),r)≤ε}∈F(I).

Since y^j → z as j → ∞, by Theorem 3.7, ϕ(Υnr(yj)−Υnr(y),r)→1 and ψ(Υnr(yj)−Υnr(y),r)→0, for all t > 0 as n → ∞. Hence, for n ∈ 𝒳,

ϕ(Υnr(z)−Υnr(y),t+r)≥limn→∞ϕ(Υnr(yj)−Υnr(y),t)∗ϕ(Υnr(yj)−Υnr(z),r) ≥1∗(1−ε)=1−ε

and

ψ(Υnr(z)−Υnr(y),t+r)≤limn→∞ψ(Υnr(yj)−Υnr(y),t)⋄ψ(Υnr(yj)−Υnr(z),r)≤0⋄ε=ε.

A particular k ∈ ℕ, take t=1k. Then

ϕ(Υnr(z)−Υnr(y),r)=limk→∞ϕ(Υnr(z)−Υnr(y),r+1k)≥1−ε

and

ψ(Υnr(z)−Υnr(y),r)=limk→∞ϕ(Υnr(z)−Υnr(y),r+1k)≤ε,

⇒ the set {n∈N:ϕ(Υnr(z)−Υnr(y),r)≥1−ε or ϕ(Υnr(z)−Υnr(y),r)≤ε}∈F(I) ⇒ y∈BzI[r,ε](Υr). Therefore, BzI[r,ε](Υr) is a closed set. □

Theorem 3.14.

Let 𝒮 be the compact subset of c(ϕ,ψ)I(Υr) such that z = (z_k) ∉ 𝒮. Then there exist two open sets 𝒰, 𝒱 in c(ϕ,ψ)I(Υr) such that 𝒮 ⊆ 𝒱, z ∈ 𝒰 and 𝒰 ∩ 𝒱 = ϕ.

Proof.

Let 𝒮 be a compact subset of c(ϕ,ψ)I(Υr) and z ∉ 𝒮. Then for any s ∈ 𝒮 we have z ≠ s. Since c(ϕ,ψ)I(Υr) is a Hausdorff space, then for some r > 0 and 0 < ε < 1 there exist two open balls U=BzI(r,ε)(Υr) and V=BsI(r,ε)(Υr) such that 𝒰 ∩ 𝒱 = ϕ. Consider the open cover Vs={BsI(r,ε)(Υr):s∈ S} of 𝒮 and 𝒮 is compact, therefore, there exists a finite subcover Ssi={BsiI(r,ε)(Υr):si∈ S and i=1,2,3,…,j} such that S⊆⋃i=1jVsi. Taking V=⋂i=1jVsi we have z ∉ 𝒮. Hence, 𝒰, 𝒱 are open sets such that 𝒮 ⊆ 𝒱 and 𝒰 ∩ 𝒱 = ϕ. □

4. Conclusion

The article deploys the notion of a regular matrix implemented on an initially non-convergent sequence to attain some finite limit. It further explores the convergence of the sequences generated after operating the regular Jordan totient operator in the setting of intuitionistic fuzzy norm via a collection of finite subsets of ℕ. We formulate novel sequence spaces c0(ϕ,ψ)I(Υr), c(ϕ,ψ)I(Υr), ℓ∞(ϕ,ψ)I(Υr) and ℓ_{∞(ϕ, ψ)}(ϒ^r) and study relations among them. Researchers’ interest can further aim at developing function spaces with the aid of a generalized infinite operator and study their properties.

(Communicated by Anatolij Dvurečenskij)

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Received: 2022-03-07

Accepted: 2022-04-27

Published Online: 2023-03-31

Published in Print: 2023-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ms-2023-0033

Keywords for this article

Jordan quotient operator; t-norm; t-conorm; Intuitionistic fuzzy norm; Jordan 𝓘-convergence; compactness; completeness

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