Interval-Valued Intuitionistic Fuzzy n-Fold Implicative Filters in Hoop Algebra

Abstract

This paper introduces the concept of interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras and explores their fundamental properties. We establish several new and equivalent characterizations, investigate their closure properties, and provide explicit algorithms for their construction and verification. Furthermore, we examine the relationships between interval-valued intuitionistic fuzzy n-fold implicative filters, interval-valued intuitionistic fuzzy filters, and classical n-fold implicative filters. The results presented here extend beyond straightforward generalizations, offering both practical tools and theoretical insights that are not previously available in the literature. The results presented here build upon earlier studies by systematically characterizing interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras. Several new equivalent conditions and computational methods are introduced, and relationships with existing filter concepts are clarified.

1. Introduction

Non-classical logic has become a fundamental framework in computer science for managing uncertain and imprecise information. Many non-classical logical systems, such as Hájek’s basic logic (BL) [1], Łukasiewicz many-valued logic (MV) [2], and monoidal t-norm based logic (MTL) [3], possess algebraic semantics founded on residuation and are analyzed within the context of residuated structures [4]. Their algebraic counterparts—BL-algebras, MV-algebras, and MTL-algebras—are constructed on lattices equipped with a residuated implication, highlighting the central role of residuation in algebraic logic. Accordingly, the study of algebraic structures with residuation is of great importance.

Hoop algebras, introduced by Bosbach as naturally ordered commutative residuated integral monoids [5,6], represent a significant class among residuated structures. They provide a unifying algebraic framework that encompasses several well-known logical and algebraic systems. Investigations, including unpublished work by Büchi and Owens and subsequent contributions, have revealed rich structural properties of hoops, with recent advances in hoop theory yielding deep structural results [7,8,9,10] and direct applications in fuzzy logic and its algebraic semantics [11]. Notably, BL-algebras, which are the algebraic semantics of Hájek’s basic logic, are special cases of hoop algebras.

Hoop algebras distinguish themselves from related structures such as BL-algebras, MV-algebras, and MTL-algebras by their naturally ordered, commutative, and integral monoid structure with a residuated implication [5,6]. This general and flexible framework allows for a unified treatment of various logical systems and does not require a lattice structure, enabling the investigation of filters and fuzzy substructures where order-theoretic constraints are relaxed. Recent structural results in hoop theory [7,8,9,10] have uncovered algebraic properties and closure behaviors—particularly regarding implicative filters—that are not always present in MV- or MTL-algebras. The technical simplicity and generality of hoops also facilitate algorithmic verification and structural analysis, making them suitable for the development of interval-valued intuitionistic fuzzy n-fold implicative filters, which offer both theoretical depth and practical applicability beyond more restrictive frameworks.

Fuzzy set theory and its generalizations have become essential for modeling uncertainty and vagueness in algebraic systems. Intuitionistic fuzzy sets, which incorporate degrees of membership and non-membership, have led to extensive studies of fuzzy substructures such as fuzzy ideals and fuzzy filters. Interval-valued intuitionistic fuzzy sets extend this by representing membership and non-membership degrees as intervals, providing a more expressive approach to uncertainty.

The interplay between hoop algebras and fuzzy logic has motivated the exploration of fuzzy filter theory in this context. Higher-order concepts such as implicative, positive implicative, and n-fold implicative filters are increasingly studied for their algebraic and logical significance. While these notions have been examined in various algebraic structures, a systematic study of interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras remains undeveloped.

The n-fold condition within the interval-valued intuitionistic fuzzy filter framework introduces structural and logical consequences by refining the classification of filters based on stability under repeated implication operations. Filters satisfying the n-fold implicative property but not the $(n-1)$-fold condition demonstrate new layers of complexity, with unique behaviors in their membership and non-membership intervals. This enables identification of filter classes with enhanced closure properties and distinct equivalence relations, absent in the standard IVIF setting. Explicit examples illustrate how the n-fold condition shapes the hierarchy and structure of filters in hoop algebras, providing substantive extensions to the IVIF framework and revealing novel phenomena in fuzzy algebraic systems.

This paper aims to fill this gap by introducing interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras and examining their fundamental properties. Equivalent characterizations are established, relationships with interval-valued intuitionistic fuzzy filters and classical n-fold implicative filters are clarified, and closure properties are investigated. The results systematically characterize these filters, presenting new equivalence theorems, computational algorithms, and structural results. Examples and closure properties further demonstrate the novelty and nontriviality of the proposed structures.

2. Preliminaries

In this section, we recall essential concepts and definitions that will be used throughout the paper. We begin by reviewing the basic properties of hoop algebras, followed by the notions of fuzzy sets and interval-valued intuitionistic fuzzy sets. These preliminaries provide the foundational framework necessary for the subsequent development of interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras.

Definition 1

([12]). An algebra $(\mathcal{H},\odot ,\to ,1)$ is said to be a hoop if $(\mathcal{H},\odot ,1)$ is a commutative monoid and the binary operation → fulfills the following conditions.

(H1) 

${\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{x}}=1$,

(H2) 

${\mathfrak{h}}_{\mathfrak{x}}\odot ({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}})={\mathfrak{h}}_{\mathfrak{y}}\odot ({\mathfrak{h}}_{\mathfrak{y}}\to {\mathfrak{h}}_{\mathfrak{x}})$,

(H3) 

${\mathfrak{h}}_{\mathfrak{x}}\to ({\mathfrak{h}}_{\mathfrak{y}}\to {\mathfrak{h}}_{\mathfrak{z}})=({\mathfrak{h}}_{\mathfrak{x}}\odot {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{z}}$ for all ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$.

Remark 1.

It is worth noting that the operations in hoop algebras often correspond to familiar logical connectors. For example, if we take $\mathcal{H}=\{0,1\}$, representing logical values “false” and “true” and define ⊙ as logical conjunction (∧), i.e., ${\mathfrak{h}}_{\mathfrak{x}}\odot {\mathfrak{h}}_{\mathfrak{y}}=min\{{\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\}$, and → as classical implication, i.e., ${\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}}=max\{1-{\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\}$, then $(\mathcal{H},\odot ,\to ,1)$ forms a hoop algebra. In this case, the hoop axioms are satisfied, and the structure captures the essence of classical propositional logic within the hoop framework. This illustrates that hoop algebras generalize familiar logical operations, making the abstract theory accessible and relevant to readers accustomed to classical logic.

Let H be a hoop. Define a relation ≤ on $\mathcal{H}$ by

$$\begin{array}{c}\hfill {\mathfrak{h}}_{\mathfrak{x}}\le {\mathfrak{h}}_{\mathfrak{y}}\iff {\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}}=1,\mathrm{for}\mathrm{all}{\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H}.\end{array}$$

(1)

Then the structure $(\mathcal{H},\le )$ is a partially ordered set.

Definition 2

([13]). Let $\mathcal{H}$ be a hoop algebra. A nonempty subset $\mathcal{S}\subseteq \mathcal{H}$ is said to be a sub-hoop of $\mathcal{H}$ provided that it is closed under the operations ⊙ and →, that is,

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H})\left(\begin{array}{c}{\mathfrak{h}}_{\mathfrak{x}}\odot {\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{S},{\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{S}\hfill \end{array}\right).\hfill \end{array}$$

(2)

In particular, every sub-hoop necessarily contains the unit element 1.

Proposition 1

([13]). Let $(\mathcal{H},\odot ,\to ,1)$ be a hoop algebra. Then, for arbitrary elements ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$, the following identities are satisfied:

1. 

$(\mathcal{H},\le )$ is a meet-semilattice with ${\mathfrak{h}}_{\mathfrak{x}}\wedge {\mathfrak{h}}_{\mathfrak{y}}={\mathfrak{h}}_{\mathfrak{x}}\odot ({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}})$,

2. 

${\mathfrak{h}}_{\mathfrak{x}}\le ({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}}$, ${\mathfrak{h}}_{\mathfrak{x}}\le {\mathfrak{h}}_{\mathfrak{y}}\to {\mathfrak{h}}_{\mathfrak{x}}$, ${\mathfrak{h}}_{\mathfrak{x}}\le ({\mathfrak{h}}_{\mathfrak{y}}\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}$,

3. 

$1\to {\mathfrak{h}}_{\mathfrak{x}}={\mathfrak{h}}_{\mathfrak{x}}$, ${\mathfrak{h}}_{\mathfrak{x}}\odot {\mathfrak{h}}_{\mathfrak{y}}\le {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}$, ${\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{x}}=1$,

4. 

if ${\mathfrak{h}}_{\mathfrak{x}}\le {\mathfrak{h}}_{\mathfrak{y}}$, then ${\mathfrak{h}}_{\mathfrak{z}}\to {\mathfrak{h}}_{\mathfrak{x}}\le {\mathfrak{h}}_{\mathfrak{z}}\to {\mathfrak{h}}_{\mathfrak{y}}$, ${\mathfrak{h}}_{\mathfrak{y}}\to {\mathfrak{h}}_{\mathfrak{z}}\le {\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{z}}$ and ${\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}}=1$,

5. 

${\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}}\le ({\mathfrak{h}}_{\mathfrak{z}}\to {\mathfrak{h}}_{\mathfrak{x}})\to ({\mathfrak{h}}_{\mathfrak{z}}\to {\mathfrak{h}}_{\mathfrak{y}})$, ${\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}}\le ({\mathfrak{h}}_{\mathfrak{y}}\to {\mathfrak{h}}_{\mathfrak{z}})\to ({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{z}})$,

6. 

${\mathfrak{h}}_{\mathfrak{x}}\to ({\mathfrak{h}}_{\mathfrak{y}}\to {\mathfrak{h}}_{\mathfrak{z}})={\mathfrak{h}}_{\mathfrak{y}}\to ({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{z}})$,

7. 

${\mathfrak{h}}_{\mathfrak{x}}\odot {\mathfrak{h}}_{\mathfrak{y}}\le {\mathfrak{h}}_{\mathfrak{z}}$ if and only if ${\mathfrak{h}}_{\mathfrak{y}}\le {\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{z}}$,

8. 

$({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}})\wedge ({\mathfrak{h}}_{\mathfrak{y}}\to {\mathfrak{h}}_{\mathfrak{z}})\le ({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{z}})$.

Definition 3

([13]). Let $\mathcal{H}$ be a hoop algebra. A nonempty subset $\mathcal{F}$ of $\mathcal{H}$ is a filter if the following requirements are satisfied.

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H})\left(\begin{array}{c}{\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{F}\Rightarrow {\mathfrak{h}}_{\mathfrak{x}}\odot {\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{F}\hfill \end{array}\right).\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H})\left(\begin{array}{c}{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{F},{\mathfrak{h}}_{\mathfrak{x}}\le {\mathfrak{h}}_{\mathfrak{y}}\Rightarrow {\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{F}\hfill \end{array}\right).\hfill \end{array}$$

(4)

Definition 4

([14]). Let $\mathcal{H}$ be a hoop algebra. A nonempty subset $\mathcal{F}\subseteq H$ is said to be an implicative filter of H whenever the following conditions are satisfied:

$$\begin{array}{cc}\hfill & 1\in \mathcal{F}\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{F})\left(\begin{array}{c}{\mathfrak{h}}_{\mathfrak{x}}\to (({\mathfrak{h}}_{\mathfrak{y}}\to {\mathfrak{h}}_{\mathfrak{z}})\to {\mathfrak{h}}_{\mathfrak{y}})\in \mathcal{F},{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{F}\Rightarrow {\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{F}\hfill \end{array}\right).\hfill \end{array}$$

(6)

It follows from (3) and (4) that $\mathcal{F}$ is closed with respect to ⊙ and is upward closed, respectively. Furthermore, a subset $\mathcal{F}$ of a hoop algebra $\mathcal{H}$ constitutes a filter of $\mathcal{H}$ precisely when condition (5) and

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{F})\left(\begin{array}{c}{\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{F},{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{F}\Rightarrow {\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{F}\hfill \end{array}\right)\hfill \end{array}$$

(7)

are satisfied.

Definition 5

([15]). Let $\mathcal{A}$ be a nonempty set. The intuitionistic fuzzy set on ${\mathcal{A}}^{\ast }$ is defined as a structure

$$\begin{array}{c}\hfill {\mathcal{A}}^{\ast }:=\{\langle {\mathfrak{h}}_{\mathfrak{x}},{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\rangle \mid {\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{A}\},\end{array}$$

(8)

where ${\mu }_{\mathcal{A}}:\mathcal{A}\to [0,1]$ is the degree of membership of ${\mathfrak{h}}_{\mathfrak{x}}$ to $\mathcal{C}$ and ${\gamma }_{\mathcal{A}}:\mathcal{A}\to [0,1]$ is the degree of non-membership of ${\mathfrak{h}}_{\mathfrak{x}}$ to $\mathcal{C}$ such that $0\le {\mu }_A\left({\mathfrak{h}}_{\mathfrak{x}}\right)+{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\le 1$, and the intuitionistic fuzzy set in (8) is simply denoted by $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$j.

Let $\mathcal{D}[0,1]$ denote the collection of all closed subintervals of the unit interval $[0,1]$. For any two elements ${\mathcal{D}}_1,{\mathcal{D}}_2\in D[0,1]$ with ${\mathcal{D}}_1=[{\mathfrak{u}}_1,{\mathfrak{v}}_1]$ and ${\mathcal{D}}_2=[{\mathfrak{u}}_2,{\mathfrak{v}}_2]$, define

$$rmin({\mathcal{D}}_1,{\mathcal{D}}_2)=[min\{{\mathfrak{u}}_1,{\mathfrak{u}}_2\},min\{{\mathfrak{v}}_1,{\mathfrak{v}}_2\}],$$

and

$$rmax({\mathcal{D}}_1,{\mathcal{D}}_2)=[max\{{\mathfrak{u}}_1,{\mathfrak{u}}_2\},max\{{\mathfrak{v}}_1,{\mathfrak{v}}_2\}].$$

More generally, for a family $\{{\mathcal{D}}_{\mathfrak{i}}=[{\mathfrak{u}}_{\mathfrak{i}},{\mathfrak{v}}_{\mathfrak{i}}]\mid \mathfrak{i}\in \mathfrak{I}\}\subseteq \mathcal{D}[0,1]$, we set

$$r\underset{\mathfrak{i}\in \mathfrak{I}}{sup}{\mathcal{D}}_{\mathfrak{i}}=\left[\underset{\mathfrak{i}\in I}{sup}{\mathfrak{u}}_{\mathfrak{i}},\underset{\mathfrak{i}\in \mathfrak{I}}{sup}{\mathfrak{v}}_{\mathfrak{i}}\right],r\underset{\mathfrak{i}\in \mathfrak{I}}{inf}{\mathcal{D}}_{\mathfrak{i}}=\left[\underset{\mathfrak{i}\in \mathfrak{I}}{inf}{\mathfrak{u}}_{\mathfrak{i}},\underset{\mathfrak{i}\in \mathfrak{I}}{inf}{\mathfrak{v}}_{\mathfrak{i}}\right].$$

Furthermore, an order relation on $\mathcal{D}[0,1]$ is introduced by declaring that ${\mathcal{D}}_1\ge {\mathcal{D}}_2$ if and only if ${\mathfrak{u}}_1\ge {\mathfrak{u}}_2$ and ${\mathfrak{v}}_1\le {\mathfrak{v}}_2$. The relations ${\mathcal{D}}_1\le {\mathcal{D}}_2$ and ${\mathcal{D}}_1={\mathcal{D}}_2$ are defined analogously.

Definition 6.

An interval-valued intuitionistic fuzzy set (IVIFS) on a nonempty set $\mathcal{X}$ is defined as an ordered pair $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$, where the mappings ${\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}}:\mathcal{X}\to \mathcal{D}[0,1]$ assign to each element ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{X}$ an interval-valued degree of membership and non-membership, respectively. For every ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{X}$, these intervals are given by

$${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\mu }_A^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)],{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[{\gamma }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)],$$

and satisfy the consistency condition

$$0\le {\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)+{\gamma }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)\le 1.$$

For convenience, the IVIFS $\mathcal{A}$ may also be represented in set-theoretic form as

$$\mathcal{A}=\{\langle {\mathfrak{h}}_{\mathfrak{x}},{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\rangle \mid {\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{X}\}.$$

The complement of an element ${\mathfrak{h}}_{\mathfrak{x}}$ in $\mathcal{A}$ is defined by

$$\overline{{\mu }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[1-{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right),1-{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)],\overline{{\gamma }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[1-{\gamma }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right),1-{\gamma }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)].$$

Definition 7.

An IVIFS $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ defined on a hoop $\mathcal{H}$ is said to be an IVIF filter of $\mathcal{H}$ whenever the conditions listed below hold for $•\in \{\odot ,\to \}$:

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H})\left(\begin{array}{c}{\mu }_{\mathcal{A}}({\mathfrak{h}}_{\mathfrak{x}}•{\mathfrak{h}}_{\mathfrak{y}})\ge rmin\{{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\}\hfill \\ {\gamma }_{\mathcal{A}}({\mathfrak{h}}_{\mathfrak{x}}•{\mathfrak{h}}_{\mathfrak{y}})\le rmax\{{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\}\hfill \end{array}\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H})\left(\begin{array}{c}{\mathfrak{h}}_{\mathfrak{x}}\le {\mathfrak{h}}_{\mathfrak{y}}\Rightarrow \left\{\begin{array}{c}{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\hfill \\ {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\hfill \end{array}\right.\hfill \end{array}\right).\hfill \end{array}$$

(10)

Theorem 1.

A IVIFS $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ is an IVIF filter of H if and only if

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H})\left(\begin{array}{c}{\mu }_{\mathcal{A}}\left(1\right)\ge {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\hfill \\ {\gamma }_{\mathcal{A}}\left(1\right)\le {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\hfill \end{array}\right).\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H})\left(\begin{array}{c}{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\ge rmin\{{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\mu }_{\mathcal{A}}({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}})\}\hfill \\ {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\le rmax\{{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}})\}\hfill \end{array}\right).\hfill \end{array}$$

(12)

Definition 8

([16]). A nonempty subset $\mathcal{F}$ of a hoop algebra $\mathcal{H}$ with $1\in \mathcal{F}$ is called an n-fold implicative filter if

$${\mathfrak{h}}_{\mathfrak{x}}\to \left(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{z}})\to {\mathfrak{h}}_{\mathfrak{y}}\right)\in \mathcal{F}\mathrm{and}{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{F}\Rightarrow {\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{F}$$

for all ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$.

Definition 9

([17]). A fuzzy set $\mu :\mathcal{H}\to [0,1]$ is said to be a fuzzy n-fold implicative filter of a hoop algebra $\mathcal{H}$ if it satisfies

$$\mu \left(1\right)\ge \mu \left({\mathfrak{h}}_{\mathfrak{x}}\right)\mathrm{for}\mathrm{all}{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H},$$

and

$$\mu \left({\mathfrak{h}}_{\mathfrak{x}}\to \left(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{z}})\to y\right)\right)\wedge \mu \left({\mathfrak{h}}_{\mathfrak{x}}\right)\le \mu \left({\mathfrak{h}}_{\mathfrak{y}}\right),\mathrm{for}\mathrm{all}{\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}.$$

3. IVIF $\mathbf{n}$-Fold Implicative Filters in Hoop Algebra

Interval-valued intuitionistic fuzzy sets (IVIFS) have become a central topic in fuzzy algebra, evolving through multiple definitions and applications across the literature. The foundational work of Atanassov and Gargov [18] introduced IVIFS as a generalization of fuzzy sets, paving the way for further developments and refinements. Subsequent studies, such as those by Gomathi Nayagam et al. [19,20], expanded the theory by exploring ranking and aggregation methods, highlighting the versatility of IVIFS in decision-making and data analysis. However, the interpretation and formalization of IVIFS have varied, leading to subtle distinctions in how membership and non-membership intervals are handled within algebraic contexts.

Our approach offers several advantages and clarifications in the study of IVIFS. Firstly, the definition we adopt is constructed to align naturally with algebraic operations and ideals, facilitating the integration of interval-valued fuzzy concepts into broader algebraic frameworks. This generalization extends classical ideal theory, providing new perspectives on structures such as Artinian and Noetherian algebras [21]. By rigorously considering both interval-valued membership and non-membership functions, our framework ensures mathematical consistency while enhancing applicability to a wider range of algebraic systems. Although our methodology is rooted in prior literature [18,19,20], it also adapts and clarifies these concepts for advanced algebraic applications, contributing to the ongoing development of IVIFS theory.

Having established the necessary preliminaries and basic properties, we now turn to the main focus of this paper. In the following section, we introduce and investigate interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras.

Let $\mathcal{H}$ be a hoop algebra. For the sake of readability, throughout this paper we use the notation ${\Delta }^n(u,v)$ instead of $(u^n\to v)\to u$ and ${\Delta }_w^n(u,v)$ instead of $w\to ((u^n\to v)\to u)$ for arbitrary elements $u,v,w\in H$.

Lemma 1.

Let $(\mathcal{H},\odot ,\to ,1)$ be a hoop algebra, and let ${\Delta }^n(x,y):=(x^n\to y)\to x$ for $x,y\in \mathcal{H}$. The following order-related properties hold:

1. 

$x\le {\Delta }^n(x,y)$ for all $x,y\in \mathcal{H}$.

2. 

If $x\le y$, then ${\Delta }^n(x,y)\le {\Delta }^n(y,x)$.

3. 

${\Delta }^n(x,y)$ is monotone in y.

4. 

For all $x,y\in \mathcal{H}$, $x\le (x^n\to y)\to x$ (see Proposition 1(4)).

Proof. 

1.

By Proposition 1(4), for all $x,y\in \mathcal{H}$, $x\le (x^n\to y)\to x={\Delta }^n(x,y)$.

2.

Suppose $x\le y$. Then $x^n\le y^n$, and since the implication operation is order-reversing in the antecedent, $x^n\to y\ge y^n\to x$. Thus, $(x^n\to y)\to x\le (y^n\to x)\to y$.

3.

Monotonicity in y follows since for $y\le y^{\prime }$, $x^n\to y\ge x^n\to y^{\prime }$, and thus ${\Delta }^n(x,y)\le {\Delta }^n(x,y^{\prime })$.

4.

This follows directly from the definition and Proposition 1(4).

   □

We have included Lemma 1 to summarize and justify all order-related properties of the ${\Delta }^n$ operation used in our proofs. By doing so, we aim to make the logical flow of the arguments transparent and facilitate easier verification and understanding for the reader.

Definition 10.

Let $\mathcal{H}$ be a hoop algebra. An IVIFS $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ on $\mathcal{H}$ is called an IVIF n-fold implicative filter of $\mathcal{H}$ if it satisfies condition (11) together with the following condition:

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H})\left(\begin{array}{c}{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\ge rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\hfill \\ {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\le rmax\{{\gamma }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\hfill \end{array}\right).\hfill \end{array}$$

(13)

To ensure the robust verification of interval-valued intuitionistic fuzzy n-fold implicative filters (IVIF n-fold implicative filters) within hoop algebras, we present a pseudocode that systematically checks whether an IVIFS $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ meets the criteria specified in Definition 10. This algorithm evaluates the required inequalities involving the n-fold implicative operation for all relevant triples of elements in $\mathcal{H}$ and provides a reliable decision procedure.

Algorithm 1 is designed to verify whether a given interval-valued intuitionistic fuzzy set $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ forms an IVIF n-fold implicative filter of a hoop algebra $\mathcal{H}$ according to Definition 10. The procedure iterates over all triples of elements ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$, computes the n-fold implicative operation ${\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})$, and checks the prescribed inequalities for the membership and non-membership intervals under the scaling parameter r.

Algorithm 1: Verification of IVIF n-Fold Implicative Filter in Hoop Algebras

For each triple, the algorithm verifies the following requirements:

$${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\ge rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\},$$

and

$${\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\le rmax\{{\gamma }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}.$$

If any of these conditions fails for any triple, the algorithm immediately returns False, indicating that $\mathcal{A}$ does not satisfy the definition. If all conditions hold for every triple, the algorithm returns True, confirming that $\mathcal{A}$ is an IVIF n-fold implicative filter of $\mathcal{H}$. This systematic approach guarantees thorough and reliable verification in accordance with Definition 10.

Example 1.

The binary operations → and ⊙ on the set $\mathcal{H}=\{0,1,2,3,4,5,6\}$ are defined by their Cayley tables shown in

Table 1. Cayley table of the binary operation →.

→	0	1	2	3	4	5	6
0	1	1	2	1	1	1	1
1	0	1	2	3	4	5	6
2	1	1	1	1	1	1	1
3	0	1	2	1	1	5	6
4	0	1	2	3	1	5	6
5	0	1	2	1	1	1	6
6	1	2	1	1	1	1	1

and

Table 2. Cayley table of the binary operation ⊙.

⊙	0	1	2	3	4	5	6
0	0	0	2	0	0	0	0
1	0	1	2	3	4	5	6
2	2	2	2	2	2	2	2
3	0	3	2	3	3	5	6
4	0	4	2	3	4	5	6
5	0	5	2	5	5	5	6
6	0	6	2	6	6	6	0

, respectively.

Let n be a positive integer and define an interval-valued intuitionistic fuzzy set $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ on $\mathcal{H}$ by

$${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[0.7,0.8],{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[0.1,0.2],\forall {\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}.$$

Consider the element

$${\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})={\mathfrak{h}}_{\mathfrak{x}}\to \left(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{z}})\to {\mathfrak{h}}_{\mathfrak{y}}\right)$$

for all ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$, where ${\mathfrak{h}}_{\mathfrak{y}}^n$ denotes the n-fold product of ${\mathfrak{h}}_{\mathfrak{y}}$ under ⊙.

According to Definition 10, we must verify

$$rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)$$

and

$$rmax\{{\gamma }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)$$

for all ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$, where

$$rmin([a_l,a_u],[b_l,b_u])=[min\{a_l,b_l\},min\{a_u,b_u\}]$$

and

$$rmax([a_l,a_u],[b_l,b_u])=[max\{a_l,b_l\},max\{a_u,b_u\}]$$

Since ${\mu }_{\mathcal{A}}$ and ${\gamma }_{\mathcal{A}}$ are constant for all elements,

$$rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}=rmin([0.7,0.8],[0.7,0.8])=[0.7,0.8]$$

$$rmax\{{\gamma }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}=rmax([0.1,0.2],[0.1,0.2])=[0.1,0.2]$$

and

$${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)=[0.7,0.8],{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)=[0.1,0.2]$$

Thus, the inequalities

$$[0.7,0.8]\le [0.7,0.8],[0.1,0.2]\ge [0.1,0.2]$$

hold for every choice of ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$.

The code provided in the Appendix A Section demonstrates that all conditions required by Definition 10 are satisfied for every $({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\in \mathcal{H}$. Consequently, $\left(\mathcal{A}\right)$ is an interval-valued intuitionistic fuzzy n-fold implicative filter of the hoop algebra $\left(\mathcal{H}\right)$.

Proposition 2.

Every IVIF n-fold implicative filter of a hoop algebra $\mathcal{H}$ is an IVIF filter of $\mathcal{H}$.

Proof. 

Let $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ be an IVIF n-fold implicative filter of $\mathcal{H}$ and let ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$. By assumption, condition (11) is satisfied. Taking ${\mathfrak{h}}_{\mathfrak{z}}=1$, we obtain

$$rmin\left\{{\mu }_{\mathcal{A}}\left(x\to {\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},1)\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\right\}\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right),$$

and

$$rmax\left\{{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\to {\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},1)\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\right\}\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right).$$

Since ${\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},1)=y$, the above inequalities reduce to

$$rmin\{{\mu }_{\mathcal{A}}({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}}),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right),$$

and

$$rmax\{{\gamma }_{\mathcal{A}}({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}}),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right).$$

Hence, $\mathcal{A}$ satisfies the defining conditions of an IVIF filter of $\mathcal{H}$, which completes the proof.    □

Theorem 2.

Let $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ be an IVIF filter of a hoop algebra $\mathcal{H}$. Then, for all ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H}$, the following statements are equivalent:

1. 

$\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ is an IVIF n-fold implicative filter of $\mathcal{H}$;

2. 

${\mu }_A\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$ and ${\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$;

3. 

${\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{x}}\right)={\mu }_{\mathcal{A}}\left(1\right)$ and ${\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{x}}\right)={\gamma }_{\mathcal{A}}\left(1\right)$;

4. 

${\mu }_{\mathcal{A}}\left(({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\right)={\mu }_{\mathcal{A}}\left(1\right)$ and ${\gamma }_{\mathcal{A}}\left(({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\right)={\gamma }_{\mathcal{A}}\left(1\right)$.

Proof. 

• (1) ⇒ (2). Assume that $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ is an IVIF n-fold implicative filter of $\mathcal{H}$. Then, condition (11) holds. By Proposition 1(3), we have ${\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})=1\to {\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})$. Hence,

  $${\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)=rmin\{{\mu }_{\mathcal{A}}(1\to {\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})),{\mu }_{\mathcal{A}}\left(1\right)\}\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),$$

  and

  $${\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)=rmax\{{\gamma }_{\mathcal{A}}(1\to {\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})),{\gamma }_{\mathcal{A}}\left(1\right)\}\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),$$

  which proves (2).
• (2) ⇒ (1). Let $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ be an IVIF filter of $\mathcal{H}$ satisfying (2). For arbitrary ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$, since $\mathcal{A}$ is an IVIF filter, we have

  $$rmin\{{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{z}}\right),{\mu }_{\mathcal{A}}({\mathfrak{h}}_{\mathfrak{z}}\to {\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}))\}\le {\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right),$$

  and

  $$rmax\{{\gamma }_{\mathcal{A}}\left(z\right),{\gamma }_{\mathcal{A}}(z\to {\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}))\}\ge {\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right).$$

  By Proposition 1(5), ${\mathfrak{h}}_{\mathfrak{x}}\le {\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})$, which implies

  $${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\le {\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\ge {\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right).$$

  Combining these inequalities with assumption (2), we obtain

  $$rmin\{{\mu }_{\mathcal{A}}\left(z\right),{\mu }_{\mathcal{A}}(z\to {\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}))\}\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),$$

  and

  $$rmax\{{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{z}}\right),{\gamma }_{\mathcal{A}}({\mathfrak{h}}_{\mathfrak{z}}\to {\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}))\}\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

  Therefore, $\mathcal{A}$ is an IVIF n-fold implicative filter of $\mathcal{H}$.
• (3) ⇒ (1). Assume that $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ is an IVIF filter of $\mathcal{H}$ satisfying (3). Then

  $$rmin\{{\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}({\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\to {\mathfrak{h}}_{\mathfrak{y}})\}\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right),$$

  and

  $$rmax\{{\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}({\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\to {\mathfrak{h}}_{\mathfrak{y}})\}\ge {\gamma }_{\mathcal{A}}\left(y\right).$$

  Using the assumption ${\mu }_{\mathcal{A}}({\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\to {\mathfrak{h}}_{\mathfrak{y}})={\mu }_{\mathcal{A}}\left(1\right)$ and ${\gamma }_{\mathcal{A}}({\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\to {\mathfrak{h}}_{\mathfrak{y}})={\gamma }_{\mathcal{A}}\left(1\right)$, we obtain

  $$rmin\{{\mu }_{\mathcal{A}}\left(1\right),{\mu }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{z}})\to {\mathfrak{h}}_{\mathfrak{y}})\}\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right),$$

  and

  $$rmax\{{\gamma }_{\mathcal{A}}\left(1\right),{\gamma }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{z}})\to {\mathfrak{h}}_{\mathfrak{y}})\}\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right).$$

  Consequently,

  $$rmin\{{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\mu }_{\mathcal{A}}({\mathfrak{h}}_{\mathfrak{x}}\to {\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}))\}\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right),$$

  and

  $$rmax\{{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}({\mathfrak{h}}_{\mathfrak{x}}\to {\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}))\}\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right),$$

  which shows that $\mathcal{A}$ is an IVIF n-fold implicative filter.
• (3) ⇒ (4). Taking ${\mathfrak{h}}_{\mathfrak{y}}=0$ in (3), we get

  $${\mu }_{\mathcal{A}}(({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})={\mu }_{\mathcal{A}}\left(1\right),{\gamma }_{\mathcal{A}}(({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})={\gamma }_{\mathcal{A}}\left(1\right),$$

  which proves (4).
• (4) ⇒ (3). By Proposition 1(3) and (4), we have

  $${\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\le {\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}}\mathrm{and}({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\le \left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)\to {\mathfrak{h}}_{\mathfrak{x}}.$$

  Thus,

  $${\mu }_{\mathcal{A}}(\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)\to {\mathfrak{h}}_{\mathfrak{x}})={\mu }_{\mathcal{A}}\left(1\right),{\gamma }_{\mathcal{A}}(\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)\to {\mathfrak{h}}_{\mathfrak{x}})={\gamma }_{\mathcal{A}}\left(1\right),$$

  and (3) follows.
• (3) ⇒ (2). For any ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H}$, using (3), we obtain

  $${\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)=rmin\{{\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right),{\mu }_{\mathcal{A}}\left(1\right)\}\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),$$

  and

  $${\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)=rmax\{{\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right),{\gamma }_{\mathcal{A}}\left(1\right)\}\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

  This completes the proof.    □

The following schematic roadmap illustrates the logical equivalence between the four statements in Theorem 2:

The schematic roadmap above not only clarifies the logical equivalence between the four statements in Theorem 2 but also underscores the technical depth and unified perspective achieved in the context of interval-valued intuitionistic fuzzy n-fold implicative filters on hoop algebras. It should be noted that these equivalences are established specifically within the framework of hoop algebras. While the methods and structural insights presented here may inspire analogous developments in related algebraic structures—such as residuated lattices, BL-algebras, or MV-algebras—direct generalization is not guaranteed and may require additional assumptions or modifications. Therefore, the scope of the results is confined to hoop algebras equipped with interval-valued intuitionistic fuzzy sets, and their applicability to broader classes remains an open avenue for future research.

Remark 2.

Let $[u_1,v_1],[u_2,v_2]\in \mathcal{D}[0,1]$ be two intervals, and recall that the order relation $[u_1,v_1]\le [u_2,v_2]$ holds if and only if $u_1\le u_2$ and $v_1\le v_2$. The $rmin$ and $rmax$ operators are defined as follows:

$$rmin([u_1,v_1],[u_2,v_2])=[min\{u_1,u_2\},min\{v_1,v_2\}],$$

$$rmax([u_1,v_1],[u_2,v_2])=[max\{u_1,u_2\},max\{v_1,v_2\}].$$

The compatibility between the interval order and these operators is established as follows: - If $[u_1,v_1]\le [u_2,v_2]$, then $rmin([u_1,v_1],[u_2,v_2])=[u_1,v_1]$ and $rmax([u_1,v_1],[u_2,v_2])=[u_2,v_2]$. - More generally, for any intervals $A,B$, $rmin(A,B)\le A$ and $rmin(A,B)\le B$, while $A\le rmax(A,B)$ and $B\le rmax(A,B)$. These properties ensure that the interval order is preserved under $rmin$ and $rmax$, and justify the inequalities used in the proofs, such as

$$rmin\{[{\mu }_{\mathcal{A}}^l\left(\eta \right),{\mu }_{\mathcal{A}}^u\left(\eta \right)],[{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)]\}\le [{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{y}}\right),{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{y}}\right)],$$

whenever $[{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{y}}\right),{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{y}}\right)]$ is greater than or equal to both intervals involved. By explicitly stating these compatibility properties, we clarify the logical steps in the transition from item one to item two, and ensure that the reasoning is transparent and rigorous.

Theorem 3.

Let $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ be an IVIFS of a hoop $\mathcal{H}$. Then the following statements hold:

1. 

$\mathcal{A}$ is an IVIF n-fold implicative filter of $\mathcal{H}$ if and only if

$$\begin{array}{c}\hfill (\forall {\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H})\left(\begin{array}{c}{\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},0)\right)\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),\hfill \\ {\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},0)\right)\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\hfill \end{array}\right).\end{array}$$

(14)

2. 

$\mathcal{A}$ is an IVIF n-fold implicative filter of $\mathcal{H}$ if and only if

$$\begin{array}{c}\hfill (\forall {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H})\left(\begin{array}{c}{\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)={\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),\hfill \\ {\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)={\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\hfill \end{array}\right).\end{array}$$

(15)

Proof. 

(1) Suppose that $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ is an IVIF n-fold implicative filter of $\mathcal{H}$. Then, by Theorem 2(2), for all ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in H$,

$${\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\mathrm{and}{\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

Taking ${\mathfrak{h}}_{\mathfrak{y}}=0$, we immediately obtain

$${\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},0)\right)\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\mathrm{and}{\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},0)\right)\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),$$

which proves the necessity.

Conversely, assume that

$${\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},0)\right)\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\mathrm{and}{\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},0)\right)\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\mathrm{for}\mathrm{all}{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}.$$

Since ${\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},0)={\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}}$, it follows that

$${\mu }_{\mathcal{A}}(({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})={\mu }_{\mathcal{A}}\left(1\right)\mathrm{and}{\gamma }_{\mathcal{A}}(({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})={\gamma }_{\mathcal{A}}\left(1\right).$$

Hence, by Theorem 2(4), $\mathcal{A}$ is an IVIF n-fold implicative filter of $\mathcal{H}$.

(2) Let $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ be an IVIF n-fold implicative filter of $\mathcal{H}$. By Theorem 2(2), we have

$${\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\mathrm{and}{\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\mathrm{for}\mathrm{all}x,y\in H.$$

On the other hand, by Proposition 1(5), $x\le {\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})$, which implies

$${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\le {\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)\mathrm{and}{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\ge {\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right).$$

Therefore,

$${\mu }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)={\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\mathrm{and}{\gamma }_{\mathcal{A}}\left({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}})\right)={\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

Conversely, the sufficiency follows directly from Theorem 2.    □

Proposition 3.

If $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ is an IVIF n-fold implicative filter of $\mathcal{H}$, then it is an IVIF $(n+1)$-fold implicative filter of $\mathcal{H}$.

Proof. 

Let $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ be an IVIF n-fold implicative filter of $\mathcal{H}$. By Proposition 1(3), we have ${\mathfrak{h}}_{\mathfrak{x}}^{n+1}\le {\mathfrak{h}}_{\mathfrak{x}}^n$ for all ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$. Hence,

$${\mathfrak{h}}_{\mathfrak{x}}^n\to 0\le {\mathfrak{h}}_{\mathfrak{x}}^{n+1}\to 0,$$

which implies

$$({\mathfrak{h}}_{\mathfrak{x}}^{n+1}\to 0)\to {\mathfrak{h}}_{\mathfrak{x}}\le ({\mathfrak{h}}_{\mathfrak{x}}^n\to 0)\to {\mathfrak{h}}_{\mathfrak{x}}={\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},0).$$

Consequently,

$${\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},0)\to {\mathfrak{h}}_{\mathfrak{x}}\le ((x^{n+1}\to 0)\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}.$$

Since $\mathcal{A}$ is an IVIF n-fold implicative filter, by Theorem 2(4),

$${\mu }_{\mathcal{A}}({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},0)\to {\mathfrak{h}}_{\mathfrak{x}})={\mu }_{\mathcal{A}}\left(1\right)\mathrm{and}{\gamma }_{\mathcal{A}}({\Delta }^n({\mathfrak{h}}_{\mathfrak{x}},0)\to {\mathfrak{h}}_{\mathfrak{x}})={\gamma }_{\mathcal{A}}\left(1\right).$$

Thus,

$${\mu }_{\mathcal{A}}(((x^{n+1}\to 0)\to x)\to x)={\mu }_{\mathcal{A}}\left(1\right)\mathrm{and}{\gamma }_{\mathcal{A}}(((x^{n+1}\to 0)\to x)\to x)={\gamma }_{\mathcal{A}}\left(1\right).$$

Again by Theorem 2(4), A is an IVIF $(n+1)$-fold implicative filter of $\mathcal{H}$.    □

Proposition 4.

If $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ is an IVIF filter of $\mathcal{H}$ such that ${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)<{\mu }_{\mathcal{A}}\left(x^{\prime }\right)$ and ${\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)>{\gamma }_{\mathcal{A}}\left(x^{\prime }\right)$ for all ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$, then $\mathcal{A}$ is an IVIF n-fold implicative filter of $\mathcal{H}$.

Proof. 

Let ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$. By Proposition 1(3), ${\mathfrak{h}}_{\mathfrak{x}}^n\le {\mathfrak{h}}_{\mathfrak{x}}$, and by Proposition 1(4), ${\mathfrak{h}}_{\mathfrak{x}}^{\prime }\le {\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }$. Hence,

$${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\le {\mu }_{\mathcal{A}}\left(x^{\prime }\right)\le {\mu }_{\mathcal{A}}\left({\left(x^n\right)}^{\prime }\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\ge {\gamma }_{\mathcal{A}}\left(x^{\prime }\right)\ge {\gamma }_{\mathcal{A}}\left({\left(x^n\right)}^{\prime }\right).$$

Since $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ is an IVIF filter, we have

$$rmin\{{\mu }_{\mathcal{A}}({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}}),{\mu }_{\mathcal{A}}\left({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\right)\}\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),$$

$$rmax\{{\gamma }_{\mathcal{A}}({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}}),{\gamma }_{\mathcal{A}}\left({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\right)\}\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

Therefore,

$${\mu }_{\mathcal{A}}({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}})\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\mathrm{and}{\gamma }_{\mathcal{A}}({\left({\mathfrak{h}}_{\mathfrak{x}}^n\right)}^{\prime }\to {\mathfrak{h}}_{\mathfrak{x}})\ge {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

By Theorem 3(1), A is an IVIF n-fold implicative filter of $\mathcal{H}$.    □

Theorem 4.

An IVIFS $A=\{[{\mu }_{\mathcal{A}}^l,{\mu }_{\mathcal{A}}^u],[{\gamma }_{\mathcal{A}}^l,{\gamma }_{\mathcal{A}}^u]\}$ in $\mathcal{H}$ is an IVIF n-fold implicative filter of $\mathcal{H}$ if and only if ${\mu }_{\mathcal{A}}^l,{\mu }_{\mathcal{A}}^u,{\gamma }_{\mathcal{A}}^l$ and ${\gamma }_{\mathcal{A}}^u$ are fuzzy n-fold implicative filters of $\mathcal{H}$.

Proof. 

Since ${\mu }_{\mathcal{A}}^l\left(1\right)\ge {\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)$, ${\mu }_{\mathcal{A}}^u\left(1\right)\ge {\mu }_{\mathcal{A}}^u\left(x\right)$, ${\gamma }_{\mathcal{A}}^l\left(1\right)\le {\gamma }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)$ and ${\gamma }_{\mathcal{A}}^u\left(1\right)\le {\gamma }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)$, ${\mu }_{\mathcal{A}}\left(1\right)\ge {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$ and ${\gamma }_{\mathcal{A}}\left(1\right)\le {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$.

• Let ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H}$. Then we have

  $$\begin{array}{ccc}{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)& =& [{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{y}}\right),{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{y}}\right)]\hfill \\ & \ge & [min\{{\mu }_{\mathcal{A}}^l\left({\Delta }_x^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)\},min\{{\mu }_{\mathcal{A}}^u\left({\Delta }_x^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}]\hfill \\ & =& rmin\{[{\mu }_{\mathcal{A}}^l\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}^u\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)],[{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)]\}\hfill \\ & =& rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\},\hfill \end{array}$$

  and

  $$\begin{array}{ccc}{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)& =& [{\gamma }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{y}}\right),{\gamma }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{y}}\right)]\hfill \\ & \le & [max\{{\gamma }_{\mathcal{A}}^l\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)\},max\left\{{\gamma }_{\mathcal{A}}^u({\mathfrak{h}}_{\mathfrak{x}}\to \left({\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}^u\left(x\right)\}\right]\hfill \\ & =& rmax\{[{\gamma }_{\mathcal{A}}^l\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}^u\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)],[{\gamma }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)]\}\hfill \\ & =& rmax\{{\gamma }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}.\hfill \end{array}$$

  Hence $A=\{[{\mu }_{\mathcal{A}}^l,{\mu }_{\mathcal{A}}^u],[{\gamma }_{\mathcal{A}}^l,{\gamma }_{\mathcal{A}}^u]\}$ is an IVIF n-fold implicative filter of $\mathcal{H}$. Conversely, assume that $A=\{[{\mu }_{\mathcal{A}}^l,{\mu }_{\mathcal{A}}^u],[{\gamma }_{\mathcal{A}}^l,{\gamma }_{\mathcal{A}}^u]\}$ is an IVIF n-fold implicative filter of $\mathcal{H}$. Let ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$. Then $[{\mu }_{\mathcal{A}}^l\left(1\right),{\mu }_{\mathcal{A}}^u\left(1\right)]={\mu }_{\mathcal{A}}\left(1\right)\ge {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)]$; hence ${\mu }_{\mathcal{A}}^l\left(1\right)\ge {\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)$ and ${\gamma }_{\mathcal{A}}^l\left(1\right)\le {\gamma }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)$. Let ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$. Then

  $$\begin{array}{ccc}[{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{y}}\right),{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{y}}\right)]& =& {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\hfill \\ & \ge & rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\}\hfill \\ & =& rmin\{[{\mu }_{\mathcal{A}}^l\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}^u\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)],[{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{y}}\right),{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{y}}\right)]\}\hfill \\ & =& [min\{{\mu }_{\mathcal{A}}^l\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{y}}\right)\},min\{{\mu }_{\mathcal{A}}^u\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{y}}\right)\}].\hfill \end{array}$$

  Hence, ${\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{y}}\right)\ge min\{{\mu }_{\mathcal{A}}^l\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}$ and ${\mu }_{\mathcal{A}}^u\left(y\right)\ge min\{{\mu }_{\mathcal{A}}^u\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}.$ Also

  $$\begin{array}{ccc}[{\gamma }_{\mathcal{A}}^l\left(y\right),{\gamma }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{y}}\right)]& =& {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\hfill \\ & \le & rmax\{{\gamma }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\hfill \\ & =& rmax\{[{\gamma }_{\mathcal{A}}^l\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}^u\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)],[{\gamma }_{\mathcal{A}}^l\left(x\right),{\gamma }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)]\}\hfill \\ & =& [max\{{\gamma }_{\mathcal{A}}^l\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)\},max\{{\gamma }_{\mathcal{A}}^u\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}].\hfill \end{array}$$

  Hence ${\gamma }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{y}}\right)\le max\{{\gamma }_{\mathcal{A}}^l({\mathfrak{h}}_{\mathfrak{x}}\to \left({\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}^l\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}$ and ${\gamma }_{\mathcal{A}}^u\left({\mathfrak{h}}_{\mathfrak{y}}\right)\le max\{{\gamma }_{\mathcal{A}}^u(x\to \left({\Delta }^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)),{\gamma }_{\mathcal{A}}^u\left(x\right)\}.$ Therefore, ${\mu }_{\mathcal{A}}^l,{\mu }_{\mathcal{A}}^u,{\gamma }_{\mathcal{A}}^l$ and ${\gamma }_{\mathcal{A}}^u$ are fuzzy n-fold implicative filters of $\mathcal{H}$.    □

Proposition 5.

If $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ and $\mathcal{B}=({\mu }_{\mathcal{B}},{\gamma }_{\mathcal{B}})$ are IVIF n-fold implicative filters of $\mathcal{H}$, then $\mathcal{A}\cap \mathcal{B}$ is an IVIF n-fold implicative filter of $\mathcal{H}$.

Proof. 

Let $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ and $\mathcal{B}=({\mu }_{\mathcal{B}},{\gamma }_{\mathcal{B}})$ be IVIF n-fold implicative filters of $\mathcal{H}$. Define $\mathcal{A}\cap \mathcal{B}=({\mu }_{\mathcal{A}\cap \mathcal{B}},{\gamma }_{\mathcal{A}\cap \mathcal{B}})$ by

$${\mu }_{\mathcal{A}\cap \mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=rmin\{{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\mu }_{\mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\},{\gamma }_{\mathcal{A}\cap \mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=rmax\{{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}.$$

Since ${\mu }_{\mathcal{A}}\left(1\right)\ge {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$ and ${\mu }_{\mathcal{B}}\left(1\right)\ge {\mu }_{\mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$, we have

$${\mu }_{\mathcal{A}\cap \mathcal{B}}\left(1\right)\ge {\mu }_{kA\cap \mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

Similarly,

$${\gamma }_{\mathcal{A}\cap \mathcal{B}}\left(1\right)\le {\gamma }_{\mathcal{A}\cap \mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

Let ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$. Since $\mathcal{A}$ and $\mathcal{B}$ are IVIF n-fold implicative filters of $\mathcal{H}$, we have

$$rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\le {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right),$$

and

$$rmin\{{\mu }_{\mathcal{B}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\le {\mu }_{\mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{y}}\right).$$

Since ${\mu }_{\mathcal{A}\cap \mathcal{B}}\left(x\right)=min\{{\mu }_{\mathcal{A}}\left(x\right),{\mu }_{\mathcal{B}}\left(x\right)\}$ for all $x\in \mathcal{H}$, it follows that

$$\begin{array}{cc}\hfill rmin\{{\mu }_{\mathcal{A}\cap \mathcal{B}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}\cap \mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}& =min\{rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\},\hfill \\ & rmin\{{\mu }_{\mathcal{B}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\}\hfill \\ & \le min\{{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right),{\mu }_{\mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\}\hfill \\ & ={\mu }_{\mathcal{A}\cap \mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{y}}\right).\hfill \end{array}$$

Similarly, since ${\gamma }_{\mathcal{A}\cap \mathcal{B}}\left(x\right)=max\{{\gamma }_{\mathcal{A}}\left(x\right),{\gamma }_{\mathcal{B}}\left(x\right)\}$ for all $x\in \mathcal{H}$, we obtain

$$rmax\{{\gamma }_{\mathcal{A}\cap \mathcal{B}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}\cap \mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\ge {\gamma }_{\mathcal{A}\cap \mathcal{B}}\left({\mathfrak{h}}_{\mathfrak{y}}\right).$$

Hence, $\mathcal{A}\cap \mathcal{B}$ is an IVIF n-fold implicative filter of $\mathcal{H}$.   □

Corollary 1.

If $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ is an IVIF n-fold implicative filter of $\mathcal{H}$, then its complement $\overline{\mathcal{A}}=(\overline{{\mu }_{\mathcal{A}}},\overline{{\gamma }_{\mathcal{A}}})$ is also an IVIF n-fold implicative filter of $\mathcal{H}$.

Definition 11.

Let $\mathcal{A}=\{\langle {\mathfrak{h}}_{\mathfrak{x}},{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\rangle :{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}\}$ be an IVIFS on $\mathcal{H}$. The operators $\oplus \mathcal{A}$ and $\otimes \mathcal{A}$ are defined by

$$\oplus \mathcal{A}=\{\langle {\mathfrak{h}}_{\mathfrak{x}},{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),\overline{{\mu }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\rangle :{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}\},$$

and

$$\otimes \mathcal{A}=\{\langle {\mathfrak{h}}_{\mathfrak{x}},\overline{{\gamma }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\rangle :{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}\}.$$

To facilitate the construction and verification of derived interval-valued intuitionistic fuzzy sets (IVIFS) through the operators ⊕ and ⊗, we provide a pseudocode that systematically generates the sets $\oplus \mathcal{A}$ and $\otimes \mathcal{A}$ from a given IVIFS $\mathcal{A}=\{\langle {\mathfrak{h}}_{\mathfrak{x}},{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\rangle :{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}\}$ on a hoop $\mathcal{H}$. This procedure computes the respective complements of the membership and non-membership intervals for each element and assembles the resulting sets in accordance with Definition 11.

Algorithm 2 describes the process for constructing the sets $\oplus \mathcal{A}$ and $\otimes \mathcal{A}$ from an interval-valued intuitionistic fuzzy set $\mathcal{A}$ on a hoop $\mathcal{H}$. For each element ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$, the algorithm computes the complement intervals $\overline{{\mu }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$ and $\overline{{\gamma }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$ as specified in Definition 11 and assembles the resulting ordered pairs into the sets $\oplus \mathcal{A}$ and $\otimes \mathcal{A}$.

This systematic approach ensures that the operators ⊕ and ⊗ are applied consistently, yielding derived IVIFS that reflect the intended structure and properties as defined.

Algorithm 2: Construction of $\oplus \mathcal{A}$ and $\otimes \mathcal{A}$ for an IVIFS on a Hoop

Example 2.

Consider the hoop algebra $(\mathcal{H},\odot ,\to ,1)$ given in Example 1 together with the interval-valued intuitionistic fuzzy set $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ defined therein.

Since $\mathcal{A}$ is an interval-valued intuitionistic fuzzy n-fold implicative filter of $\mathcal{H}$ (see Example 1), it satisfies condition (11) as well as inequality (13). By Definition 11, the IVIFS’s $\oplus \mathcal{A}$ and $\otimes \mathcal{A}$ are obtained by interchanging the membership and non-membership functions. Consequently, the defining inequalities in Definition 10 remain valid for both $\oplus \mathcal{A}$ and $\otimes \mathcal{A}$.

Therefore, $\oplus \mathcal{A}$ and $\otimes \mathcal{A}$ satisfy condition (13) of Definition 10, and hence they are interval-valued intuitionistic fuzzy n-fold implicative filters of the hoop algebra $\mathcal{H}$.

Theorem 5.

Let $\mathcal{A}=\{\langle {\mathfrak{h}}_{\mathfrak{x}},{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\rangle \mid {\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}\}$ be an IVIF n-fold implicative filter of a hoop algebra $\mathcal{H}$. Then both $\oplus \mathcal{A}$ and $\otimes \mathcal{A}$ are IVIF n-fold implicative filters of $\mathcal{H}$.

Proof. 

Assume that $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ is an IVIF n-fold implicative filter of $\mathcal{H}$.

(i)    The case of $\oplus \mathcal{A}$. For any ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$, since ${\mu }_{\mathcal{A}}\left(1\right)\ge {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$, we have

$$\overline{{\mu }_{\mathcal{A}}}\left(1\right)=1-{\mu }_{\mathcal{A}}\left(1\right)\le 1-{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=\overline{{\mu }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

Let ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$. Then

$$\begin{array}{cc}\hfill \overline{{\mu }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)& =1-{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\hfill \\ & \ge 1-rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\hfill \\ & =max\{1-r{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),1-r{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}.\hfill \end{array}$$

Hence $\oplus \mathcal{A}$ satisfies the defining conditions of an IVIF n-fold implicative filter.

(ii)    The case of $\otimes \mathcal{A}$. For any ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$, since ${\gamma }_{\mathcal{A}}\left(1\right)\le {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$, we obtain

$$\overline{{\gamma }_{\mathcal{A}}}\left(1\right)=1-{\gamma }_{\mathcal{A}}\left(1\right)\ge 1-{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=\overline{{\gamma }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

Let ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$. Then

$$\begin{array}{cc}\hfill \overline{{\gamma }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)& =1-{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\hfill \\ & \ge 1-rmax\{{\gamma }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\hfill \\ & =min\{1-r{\gamma }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),1-r{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}.\hfill \end{array}$$

Thus $\otimes \mathcal{A}$ is also an IVIF n-fold implicative filter of $\mathcal{H}$.    □

Example 3.

Let $\mathcal{H}=\{0,1,2,3,4,5,6\}$ be the hoop algebra with operations ⊙ and → as in Example 1. Define the interval-valued intuitionistic fuzzy set $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ by

$${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[0.7,0.8],{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[0.1,0.2],\forall {\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}.$$

As shown in Example 1, $\mathcal{A}$ is an IVIF n-fold implicative filter of $\mathcal{H}$.

Now, consider the fuzzy complement sets:

$$\oplus \mathcal{A}=(\overline{{\mu }_{\mathcal{A}}},{\gamma }_{\mathcal{A}}),\mathit{where}\overline{{\mu }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[1-0.8,1-0.7]=[0.2,0.3],$$

$$\otimes \mathcal{A}=({\mu }_{\mathcal{A}},\overline{{\gamma }_{\mathcal{A}}}),\mathit{where}\overline{{\gamma }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[1-0.2,1-0.1]=[0.8,0.9],$$

for all ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$.

Let us verify that $\oplus \mathcal{A}$ and $\otimes \mathcal{A}$ are IVIF n-fold implicative filters of $\mathcal{H}$.

• Case 1: $\oplus \mathcal{A}$. Since $\overline{{\mu }_{\mathcal{A}}}$ is constant, for any ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$,

  $$\overline{{\mu }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[0.2,0.3],\overline{{\mu }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)=[0.2,0.3].$$

  For any interval-valued operation,

  $$rmin\{\overline{{\mu }_{\mathcal{A}}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),\overline{{\mu }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}=[0.2,0.3],$$

  so

  $$[0.2,0.3]\le [0.2,0.3]$$

  holds for all ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}$.
• Case 2: $\otimes \mathcal{A}$. Since $\overline{{\gamma }_{\mathcal{A}}}$ is constant,

  $$\overline{{\gamma }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[0.8,0.9],\overline{{\gamma }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)=[0.8,0.9].$$

  For any interval-valued operation,

  $$rmax\{\overline{{\gamma }_{\mathcal{A}}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),\overline{{\gamma }_{\mathcal{A}}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}=[0.8,0.9],$$

  so

  $$[0.8,0.9]\ge [0.8,0.9]$$

  holds for all ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}$.
• Thus, both $\oplus \mathcal{A}$ and $\otimes \mathcal{A}$ are IVIF n-fold implicative filters of $\mathcal{H}$, as required by the theorem.

Theorem 6.

An IVIFS $\mathcal{A}=\{\langle x,{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\rangle :{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}\}$ is an IVIF n-fold implicative filter of $\mathcal{H}$ if and only if for every $[{\mathfrak{e}}_1,{\mathfrak{e}}_2],[{\mathfrak{f}}_1,{\mathfrak{f}}_2]\in \mathcal{D}[0,1]$, the sets $U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])$ and $L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2])$ are either empty or n-fold implicative filters of $\mathcal{H}$.

Proof. 

(⇒) Let $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ be an IVIF n-fold implicative filter of $\mathcal{H}$ and let $[{\mathfrak{f}}_1,{\mathfrak{f}}_2],[{\mathfrak{e}}_1,{\mathfrak{e}}_2]\in \mathcal{D}[0,1]$ such that $U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])\ne \varnothing$ and $L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2])\ne \varnothing$.

Since ${\mu }_{\mathcal{A}}\left(1\right)\ge [{\mathfrak{e}}_1,{\mathfrak{e}}_2]$ and ${\gamma }_{\mathcal{A}}\left(1\right)\le [{\mathfrak{f}}_1,{\mathfrak{f}}_2]$, we have

$$1\in U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])\cap L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2]).$$

Let ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$ such that ${\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}),{\mathfrak{h}}_{\mathfrak{x}}\in U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])$.

Then

$${\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)\ge {\mathfrak{e}}_1\mathrm{and}{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\ge {\mathfrak{e}}_1.$$

Since $\mathcal{A}$ is an IVIF n-fold implicative filter, we have

$${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\ge min\{{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\ge [{\mathfrak{e}}_1,{\mathfrak{e}}_2],$$

which implies that ${\mathfrak{h}}_{\mathfrak{y}}\in U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])$. Hence, $U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])$ is an n-fold implicative filter of $\mathcal{H}$.

Similarly, let ${\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}),{\mathfrak{h}}_{\mathfrak{x}}\in L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2])$. Then

$${\gamma }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)\le [{\mathfrak{f}}_1,{\mathfrak{f}}_2]\mathrm{and}{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\le [{\mathfrak{f}}_1,{\mathfrak{f}}_2].$$

Since $\mathcal{A}$ is an IVIF n-fold implicative filter, we obtain

$${\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\le max\{{\gamma }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}\le [{\mathfrak{f}}_1,{\mathfrak{f}}_2].$$

Therefore, ${\mathfrak{h}}_{\mathfrak{y}}\in L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2])$, and hence $L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2])$ is an n-fold implicative filter of $\mathcal{H}$.

(⇐) Assume that every nonempty set $U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])$ and $L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2])$ is an n-fold implicative filter of $\mathcal{H}$.

If ${\mu }_{\mathcal{A}}\left(1\right)\ge {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$ does not hold for some ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$, then there exists ${{\mathfrak{h}}_{\mathfrak{x}}}_0\in \mathcal{H}$ such that ${\mu }_{\mathcal{A}}\left(1\right)<{\mu }_{\mathcal{A}}\left({{\mathfrak{h}}_{\mathfrak{x}}}_0\right)$. Choose

$$[{\mathfrak{e}}_1,{\mathfrak{e}}_2]={\displaystyle \frac{1}{2}}\left({\mu }_{\mathcal{A}}\left(1\right)+{\mu }_{\mathcal{A}}\left({{\mathfrak{h}}_{\mathfrak{x}}}_0\right)\right).$$

Then ${{\mathfrak{h}}_{\mathfrak{x}}}_0\in U({\mu }_{\mathcal{A}},[t_1,t_2])$, so $U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])\ne \varnothing$. Since it is a filter, $1\in U({\mu }_{\mathcal{A}},[t_1,t_2])$, which yields ${\mu }_{\mathcal{A}}\left(1\right)\ge [{\mathfrak{e}}_1,{\mathfrak{e}}_2]$, a contradiction. Hence ${\mu }_{\mathcal{A}}\left(1\right)\ge {\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$ for all ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$.

A similar argument shows that ${\gamma }_{\mathcal{A}}\left(1\right)\le {\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)$ for all ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$.

Now suppose that

$${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)\ge rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}.$$

does not hold for some ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in H$. Then there exist $u_0,v_0,w_0\in \mathcal{H}$ such that

$${\mu }_{\mathcal{A}}\left(v_0\right)<rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{u_0}^n(v_0,w_0)\right),{\mu }_{\mathcal{A}}\left(u_0\right)\}.$$

Choose

$$[p_1,p_2]={\displaystyle \frac{1}{2}}\left({\mu }_{\mathcal{A}}\left(v_0\right)+rmin\{{\mu }_{\mathcal{A}}\left({\Delta }_{u_0}^n(v_0,w_0)\right),{\mu }_{\mathcal{A}}\left(u_0\right)\}\right).$$

Then ${\Delta }_{u_0}^n(v_0,w_0),u_0\in U({\mu }_{\mathcal{A}},[p_1,p_2])$ but $v_0\notin U({\mu }_{\mathcal{A}},[p_1,p_2])$, contradicting the filter property. Hence the inequality holds.

An analogous argument shows that

$${\gamma }_{\mathcal{A}}\left(y\right)\le rmax\{{\gamma }_{\mathcal{A}}\left({\Delta }_x^n(y,z)\right),{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}$$

for all ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$. Therefore, A is an IVIF n-fold implicative filter of $\mathcal{H}$.    □

Theorem 7.

Let $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ be an IVIF n-fold implicative filter of $\mathcal{H}$. Then the following equalities hold:

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H})\left(\begin{array}{c}{\mu }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}})={\mu }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})\hfill \\ {\gamma }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}})={\gamma }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})\hfill \end{array}\right).\hfill \end{array}$$

(16)

Proof. 

Let $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ be an IVIF n-fold implicative filter of $\mathcal{H}$. By Theorem 6, for any $[{\mathfrak{f}}_1,{\mathfrak{f}}_2],[{\mathfrak{e}}_1,{\mathfrak{e}}_2]\in \mathcal{D}[0,1]$, the nonempty sets $U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])$ and $L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2])$ are n-fold implicative filters of $\mathcal{H}$.

Let

$$[{\mathfrak{e}}_1,{\mathfrak{e}}_2]={\mu }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}}\right)\mathrm{and}[{\mathfrak{f}}_1,{\mathfrak{f}}_2]={\gamma }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}}\right).$$

Then

$$({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}}\in U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])\cap L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2]).$$

By Proposition 1(2), we have ${\mathfrak{h}}_{\mathfrak{y}}^n\le ({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}$. Moreover, by Proposition 1(4),

$$({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}}\le ({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to ({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}}).$$

Hence,

$$({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to \left(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\right)\in U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])\cap L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2]).$$

Similarly, since ${\mathfrak{h}}_{\mathfrak{x}}\le ({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}$, Proposition 1(4) yields

$$(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{y}}\le {\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}},$$

and consequently,

$$({\mathfrak{h}}_{\mathfrak{x}}\to {\mathfrak{h}}_{\mathfrak{y}})\to (({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})\le ((({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{y}})\to (({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}).$$

Thus,

$$((({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{y}})\to (({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})\in U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])\cap L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2]).$$

Since $U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])$ and $L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2])$ are n-fold implicative filters, Proposition 1(3) implies

$$1\to \left(((({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{y}})\to (({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})\right)\in U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])\cap L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2]).$$

Furthermore, $1\in U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])\cap L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2])$ and $({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\in U({\mu }_{\mathcal{A}},[{\mathfrak{e}}_1,{\mathfrak{e}}_2])\cap L({\gamma }_{\mathcal{A}},[{\mathfrak{f}}_1,{\mathfrak{f}}_2])$. Hence,

$$[{\mathfrak{e}}_1,{\mathfrak{e}}_2]\le {\mu }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\right)\mathrm{and}[{\mathfrak{f}}_1,{\mathfrak{f}}_2]\ge {\gamma }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\right).$$

Therefore,

$${\mu }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}}\right)\le {\mu }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\right){\mu }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\right),$$

and

$${\gamma }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}}\right)\ge {\gamma }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\right).$$

By symmetry, the reverse inequalities also hold, and hence

$${\mu }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}}\right)={\mu }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\right),$$

and

$${\gamma }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}}\right)={\gamma }_{\mathcal{A}}\left(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}\right).$$

   □

Example 4.

Consider the hoop algebra $\mathcal{H}=\{0,1,2,3,4,5,6\}$ with the binary operations ⊙ and → as described in Example 1. Let n be any positive integer and define the interval-valued intuitionistic fuzzy set $\mathcal{A}=({\mu }_{\mathcal{A}},{\gamma }_{\mathcal{A}})$ on $\mathcal{H}$ by

$${\mu }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[0.7,0.8],{\gamma }_{\mathcal{A}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)=[0.1,0.2],\forall {\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}.$$

As shown in Example 1, $\mathcal{A}$ is an IVIF n-fold implicative filter of $\mathcal{H}$.

Let us verify the equalities in Theorem 7 for arbitrary ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H}$:

$${\mu }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}})={\mu }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}),$$

and

$${\gamma }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}})={\gamma }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}}).$$

Since ${\mu }_{\mathcal{A}}$ and ${\gamma }_{\mathcal{A}}$ are constant functions,

$${\mu }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}})=[0.7,0.8],{\mu }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})=[0.7,0.8],$$

$${\gamma }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{x}}^n\to {\mathfrak{h}}_{\mathfrak{y}})\to {\mathfrak{h}}_{\mathfrak{y}})=[0.1,0.2],{\gamma }_{\mathcal{A}}(({\mathfrak{h}}_{\mathfrak{y}}^n\to {\mathfrak{h}}_{\mathfrak{x}})\to {\mathfrak{h}}_{\mathfrak{x}})=[0.1,0.2].$$

Therefore, for all ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{H}$, the equalities in Theorem 7 are satisfied.

This example demonstrates that the equalities in Theorem 7 hold for the IVIF n-fold implicative filter defined in Example 1.

For any fixed interval numbers ${\tilde{\mathfrak{r}}}^{+},{\tilde{\mathfrak{r}}}^{-},{\tilde{\mathfrak{s}}}^{+},{\tilde{\mathfrak{s}}}^{-}\in \left[[0,1]\right]$ such that ${\tilde{\mathfrak{r}}}^{+}\succ {\tilde{\mathfrak{r}}}^{-}$, ${\tilde{\mathfrak{s}}}^{+}\succ {\tilde{\mathfrak{s}}}^{-}$ and a nonempty subset G of $\mathcal{H}$, the IVIS

$${\mathcal{A}}^G\left[\begin{array}{cc}{\tilde{\mathfrak{r}}}^{+},& {\tilde{\mathfrak{s}}}^{-}\\ {\tilde{\mathfrak{r}}}^{-},& {\tilde{\mathfrak{s}}}^{+}\end{array}\right]=\left(\mathcal{H},{\mu }_{\mathcal{A}}^G\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right],{\gamma }_{\mathcal{A}}^G\left[\begin{array}{c}{\tilde{\mathfrak{s}}}^{-}\\ {\tilde{\mathfrak{s}}}^{+}\end{array}\right]\right)$$

in $\mathcal{H}$, where

$${\mu }_{\mathcal{A}}^G\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left({\mathfrak{h}}_{\mathfrak{x}}\right)=\left\{\begin{array}{cc}{\tilde{\mathfrak{r}}}^{+}& \mathrm{if}{\mathfrak{h}}_{\mathfrak{x}}\in G\\ {\tilde{\mathfrak{r}}}^{-}& otherwise,\end{array}\right.$$

$${\gamma }_{\mathcal{A}}^G\left[\begin{array}{c}{\tilde{\mathfrak{s}}}^{-}\\ {\tilde{\mathfrak{s}}}^{+}\end{array}\right]\left({\mathfrak{h}}_{\mathfrak{x}}\right)=\left\{\begin{array}{cc}{\tilde{\mathfrak{s}}}^{-}& \mathrm{if}{\mathfrak{h}}_{\mathfrak{x}}\in G\\ {\tilde{\mathfrak{s}}}^{+}& otherwise.\end{array}\right.$$

Lemma 2.

If the constant 1 of $\mathcal{H}$ is in a nonempty subset $\mathcal{G}$ of $\mathcal{H}$, then the IVIFS

$${\mathcal{A}}^{\mathcal{G}}=\left(\mathcal{H},{\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right],{\gamma }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{s}}}^{-}\\ {\tilde{\mathfrak{s}}}^{+}\end{array}\right]\right)$$

satisfies the conditions (11).

Proof. 

If $1\in \mathcal{G}$, then ${\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left(1\right)={\tilde{\mathfrak{r}}}^{+}$, and ${\gamma }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{s}}}^{-}\\ {\tilde{\mathfrak{s}}}^{+}\end{array}\right]\left(1\right)={\tilde{\mathfrak{s}}}^{-}$. Thus,

$$\begin{array}{cc}\hfill & (\forall {\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H})\left(\begin{array}{c}{\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left(1\right)={\tilde{\mathfrak{r}}}^{+}⪰{\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left({\mathfrak{h}}_{\mathfrak{x}}\right)\hfill \\ {\gamma }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{s}}}^{-}\\ {\tilde{\mathfrak{s}}}^{+}\end{array}\right]\left(1\right)={\tilde{\mathfrak{s}}}^{-}⪯{\gamma }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{s}}}^{-}\\ {\tilde{\mathfrak{s}}}^{+}\end{array}\right]\left({\mathfrak{h}}_{\mathfrak{x}}\right)\hfill \end{array}\right).\hfill \end{array}$$

(17)

Hence ${\mathcal{A}}^G\left[\begin{array}{cc}{\tilde{\mathfrak{r}}}^{+},& {\tilde{\mathfrak{s}}}^{-}\\ {\tilde{\mathfrak{r}}}^{-},& {\tilde{\mathfrak{s}}}^{+}\end{array}\right]$ satisfies the conditions (11).    □

Lemma 3.

If the IVIS

$${\mathcal{A}}^{\mathcal{G}}=\left(\mathcal{H},{\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right],{\gamma }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{s}}}^{-}\\ {\tilde{\mathfrak{s}}}^{+}\end{array}\right]\right)$$

in $\mathcal{H}$ satisfies the condition (11), then the constant 1 of $\mathcal{H}$ is in the nonempty subset $\mathcal{G}$ of $\mathcal{H}$.

Proof. 

Assume that ${\mathcal{A}}^{\mathcal{G}}$ satisfies the condition (11). Then, for all ${\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{H}$,

$${\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left(1\right)⪰{\mu }_{\mathcal{A}}^G\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

Since $\mathcal{G}$ is nonempty, there exists $\mathfrak{g}\in \mathcal{G}$. By the definition of ${\mu }_{\mathcal{A}}^{\mathcal{G}}$,

$${\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left(\mathfrak{g}\right)={\tilde{\mathfrak{r}}}^{+}.$$

Hence,

$${\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left(1\right)⪰{\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left(\mathfrak{g}\right)={\tilde{\mathfrak{r}}}^{+}⪰{\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left(1\right),$$

which implies

$${\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left(1\right)={\tilde{\mathfrak{r}}}^{+}.$$

Therefore, $1\in \mathcal{G}$.    □

Theorem 8.

The IVIS

$${\mathcal{A}}^{\mathcal{G}}=\left(\mathcal{H},{\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right],{\gamma }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{s}}}^{-}\\ {\tilde{\mathfrak{s}}}^{+}\end{array}\right]\right)$$

in $\mathcal{H}$ is an IVIF n-fold implicative filter of $\mathcal{H}$ if and only if the nonempty subset $\mathcal{G}$ of $\mathcal{H}$ is an n-fold implicative filter of $\mathcal{H}$.

Proof. 

Assume that ${\mathcal{A}}^{\mathcal{G}}$ is an IVIF n-fold implicative filter of $\mathcal{H}$. Since ${\mathcal{A}}^{\mathcal{G}}$ satisfies condition (17), by Lemma 3, $1\in \mathcal{G}$.

Let ${\mathfrak{h}}_{\mathfrak{x}},{\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}\in \mathcal{H}$ with ${\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}),{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{G}$. Then

$${\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)={\tilde{a}}^{+}={\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left({\mathfrak{h}}_{\mathfrak{x}}\right).$$

Hence,

$${\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{a}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left({\mathfrak{h}}_{\mathfrak{y}}\right)⪰rmin\left\{{\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}^{\mathcal{G}}\left[\begin{array}{c}{\tilde{\mathfrak{r}}}^{+}\\ {\tilde{\mathfrak{r}}}^{-}\end{array}\right]\left({\mathfrak{h}}_{\mathfrak{x}}\right)\right\}={\tilde{\mathfrak{r}}}^{+}.$$

Thus ${\mu }_{\mathcal{A}}^{\mathcal{G}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)={\tilde{\mathfrak{r}}}^{+}$, so $y\in G$. Therefore, $\mathcal{G}$ is an n-fold implicative filter of $\mathcal{H}$.

Conversely, assume that $\mathcal{G}$ is an n-fold implicative filter of $\mathcal{H}$. Since $1\in \mathcal{G}$, by Lemma 2, ${\mathcal{A}}^{\mathcal{G}}$ satisfies condition (17).

Case 1: Suppose ${\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}}),{\mathfrak{h}}_{\mathfrak{x}}\in \mathcal{G}$. Then

$${\mu }_{\mathcal{A}}^{\mathcal{G}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)={\mu }_{\mathcal{A}}^{\mathcal{G}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)={\tilde{\mathfrak{r}}}^{+},{\gamma }_{\mathcal{A}}^{\mathcal{G}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)={\gamma }_{\mathcal{A}}^{\mathcal{G}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)={\tilde{\mathfrak{s}}}^{-}.$$

Since $\mathcal{G}$ is an n-fold implicative filter, ${\mathfrak{h}}_{\mathfrak{y}}\in \mathcal{G}$, hence

$${\mu }_{\mathcal{A}}^{\mathcal{G}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)={\tilde{\mathfrak{r}}}^{+},{\gamma }_{\mathcal{A}}^{\mathcal{G}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)={\tilde{\mathfrak{s}}}^{-}.$$

Case 2: Suppose ${\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\notin \mathcal{G}$ or ${\mathfrak{h}}_{\mathfrak{x}}\notin \mathcal{G}$. Then

$${\mu }_{\mathcal{A}}^G\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)={\tilde{\mathfrak{r}}}^{-}\mathrm{or}{\mu }_{\mathcal{A}}^{\mathcal{G}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)={\tilde{a}}^{-},$$

and

$${\gamma }_{\mathcal{A}}^{\mathcal{G}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right)={\tilde{\mathfrak{s}}}^{+}\mathrm{or}{\gamma }_{\mathcal{A}}^{\mathcal{G}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)={\tilde{\mathfrak{s}}}^{+}.$$

Hence,

$$rmin\{{\mu }_{\mathcal{A}}^{\mathcal{G}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\mu }_{\mathcal{A}}^{\mathcal{G}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}={\tilde{\mathfrak{r}}}^{-},rmax\{{\gamma }_{\mathcal{A}}^{\mathcal{G}}\left({\Delta }_{{\mathfrak{h}}_{\mathfrak{x}}}^n({\mathfrak{h}}_{\mathfrak{y}},{\mathfrak{h}}_{\mathfrak{z}})\right),{\gamma }_{\mathcal{A}}^{\mathcal{G}}\left({\mathfrak{h}}_{\mathfrak{x}}\right)\}={\tilde{\mathfrak{s}}}^{+}.$$

Therefore,

$${\mu }_{\mathcal{A}}^{\mathcal{G}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)⪰{\tilde{\mathfrak{r}}}^{-},{\gamma }_{\mathcal{A}}^{\mathcal{G}}\left({\mathfrak{h}}_{\mathfrak{y}}\right)⪯{\tilde{\mathfrak{s}}}^{+}.$$

Thus, ${\mathcal{A}}^{\mathcal{G}}$ is an IVIF n-fold implicative filter of $\mathcal{H}$.    □

4. Conclusions

In this paper, we have established several equivalent characterizations of interval-valued intuitionistic fuzzy n-fold implicative filters in finite hoop algebras. The main technical contributions include the equivalence between the n-fold implicative filter property and explicit interval conditions involving the ${\Delta }^n$ operation, a schematic roadmap clarifying logical dependencies and a proof structure (see Remark 2), computational algorithms for explicit verification in finite cases, and closure properties under standard constructions such as intersection, complement, and derived operators. These results unify and extend classical fuzzy filter theory, demonstrating that interval-valued intuitionistic fuzzy filters and their n-fold counterparts can be systematically characterized and verified.

Our study is currently restricted to finite hoop algebras and interval-valued intuitionistic fuzzy sets with constant or piecewise-defined intervals. The structural properties and technical advantages of hoop algebras, as previously established in [5,6,7,8,9,10], provide a robust foundation for these results. Extending these findings to infinite algebras, more general residuated structures, or dynamic fuzzy systems remains an open challenge. Computational verification is feasible only for finite cases, and further research is needed for algorithmic approaches in infinite settings.

The findings suggest potential applications in logical systems and decision-making contexts where uncertainty and iterative implication are central. Future work may include exploring homomorphic images, categorical properties, and dynamic versions of interval-valued intuitionistic fuzzy filters.

By introducing interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras, we have extended several existing notions of implicative-type filters within this framework. Fundamental properties and characterizations have been established, and the relationships between interval-valued intuitionistic fuzzy implicative filters and their n-fold counterparts have been clarified. Several known results regarding fuzzy and intuitionistic fuzzy filters are shown to be special cases of our main findings, enriching the theory of hoop algebras and providing a unified perspective for higher-order implicative structures under interval-valued intuitionistic fuzziness. The included examples illustrate the nontriviality and applicability of the introduced concepts, further supported by the rich structural results in hoop theory [7,8,9,10].

The results open several avenues for further research, such as investigating interval-valued intuitionistic fuzzy n-fold implicative filters in more general algebraic structures like residuated lattices, BL-algebras, or related logical algebras, and studying homomorphic images and categorical properties. Applications in logical systems and decision-making problems, where uncertainty and graded implication are significant, as well as dynamic or parameterized versions of interval-valued intuitionistic fuzzy filters, are promising directions for future exploration.

The proposed structures based on interval-valued intuitionistic fuzzy sets (IVIFS) are inherently designed to handle uncertainty and graded implication, as highlighted in previous literature [19,20]. In decision-making contexts, uncertainty evolves over time, requiring models that accommodate dynamic changes in information. The IVIFS framework, with interval-valued membership and non-membership functions, is well-suited for such scenarios: intervals can be updated or recalibrated as new data becomes available, reflecting the temporal evolution of uncertainty.

For instance, in multi-criteria decision-making problems, the algorithm by Gomathi Nayagam et al. [20] shows how IVIFS can aggregate and rank alternatives under changing information. Our proposed algorithm can be adapted to time-varying uncertainty by allowing the intervals associated with each criterion or logical proposition to be functions of time or sequential observations. This enables the model to capture both the current state and progression of uncertainty, making it applicable to real-world problems such as forecasting, adaptive control, or dynamic resource allocation.

Consider a scenario where the reliability of information sources fluctuates over time. By representing membership and non-membership degrees as evolving intervals, the IVIFS-based algorithm can continually update the assessment of alternatives, providing robust solutions under time-varying uncertainty.

In summary, this paper introduces and systematically characterizes interval-valued intuitionistic fuzzy n-fold implicative filters in hoop algebras, providing explicit equivalence theorems, computational verification algorithms, and closure properties. The proposed framework extends classical fuzzy filter theory, offering new tools for managing uncertainty and graded implication in algebraic structures. Future research may explore generalizations to infinite algebras, residuated lattices, and dynamic fuzzy systems, as well as applications in decision-making and logical systems where interval-valued uncertainty is essential.