Interval Linguistic-Valued Intuitionistic Fuzzy Concept Lattice and Its Application to Linguistic Association Rule Extraction

Abstract

In a world rich with linguistic-valued data, traditional methods often lead to significant information loss when converting such data into other formats. This paper presents a novel approach for constructing an interval linguistic-valued intuitionistic fuzzy concept lattice, which adeptly manages qualitative linguistic information by leveraging the strengths of interval-valued intuitionistic fuzzy sets to represent both fuzziness and uncertainty. First, the interval linguistic-valued intuitionistic fuzzy concept lattice is constructed by integrating interval intuitionistic fuzzy sets, capturing the bidirectional fuzzy linguistic information between objects, which encompasses both positive and negative aspects. Second, by analyzing the expectations of concept extent relative to intent, and considering both the membership and non-membership perspectives of linguistic expressions, we focus on the extraction of linguistic association rules. Finally, comparative analyses and examples demonstrate the effectiveness of the proposed approach, showcasing its potential to advance the management of linguistic data in various domains.

1. Introduction

Formal concept analysis (FCA), introduced by Wille in 1982 [1], is a mathematical framework designed to systematically analyze relationships between objects and attributes in a formal context. FCA typically operates on binary data tables, which consist of an object set, an attribute set, and binary relations between them. FCA outputs include structures such as concept lattices [2], attribute implications [3], attribute topology [4], and partial order structures [5,6], allowing it to be adapted to various downstream tasks based on specific analytical requirements.

Due to the increasing diversity and complexity of data, individuals often encounter substantial uncertainty in knowledge discovery. To better express this uncertainty and the inherent fuzziness of data, Zadeh [7] proposed the concept of the fuzzy set (FS) in 1965. Fuzzy sets have since found widespread application in various fields, including economic management, science and technology, multiple objective optimization, and medical diagnosis [8,9,10,11,12]. However, one of the limitations of traditional fuzzy sets is their inability to effectively describe the degree of negation or uncertainty in the evaluation information provided by decision-makers. To address this limitation, Atanassov [13] extended the concept of fuzzy sets by proposing intuitionistic fuzzy sets (IFSs), which introduced a non-membership function in addition to the traditional membership function. This advancement allowed IFSs to simultaneously account for both positive and negative evaluations from decision-makers while implicitly considering their hesitation. The introduction of IFSs significantly enhanced the ability to handle uncertainty in decision-making processes, leading to widespread recognition and adoption across the field. Numerous scholars have since explored the theoretical foundations and practical applications of IFSs [14,15,16].

While IFSs have been effective across a range of domains, their limitation lies in the reliance on real-number evaluations for membership and non-membership degrees. Given the complexity of decision-making processes and the inherent ambiguity in human reasoning, it is often impractical to express evaluations solely as precise numerical values. In response, Atanassov [17] introduced interval-valued intuitionistic fuzzy sets (IVIFSs), where both membership and non-membership degrees are represented by intervals. This development provided a more flexible and expressive framework for capturing the inherent uncertainty and variability of real-world scenarios.

In real-world scenarios, people often express qualitative information using linguistic values. To model such linguistic expressions, many scholars have employed fuzzy linguistic approaches [18,19]. Recognizing the natural language characteristics of linguistic expressions, Xu et al. [20] introduced the concepts of linguistic truth-valued lattice implication algebra and linguistic truth-valued propositional logic. Additionally, Liao et al. [21] proposed the hesitant fuzzy linguistic term set and developed methods to measure the distance and similarity between different hesitant fuzzy linguistic term sets, providing further tools for linguistic data analysis.

In classical FCA, the relationships between objects and attributes are typically represented in binary form (i.e., 0 or 1), reflecting a crisp setting. However, many attributes in real life exhibit fuzziness, which binary representations fail to capture. To address this limitation, linguistic expressions can be used to qualitatively describe such attributes. For example, in medical consultations, a patient’s condition might be described using terms such as “serious”, “not serious”, or “very serious”. The challenge of embedding expressions into FCA has thus emerged as a key research topic [22,23]. Building on the foundation of linguistic truth-valued lattice implication algebra and linguistic term sets, Zou et al. [24,25,26] extended the concept lattice to include linguistic concept lattices, linguistic truth-valued intuitionistic fuzzy lattices, and intuitionistic fuzzy linguistic concept lattices. By incorporating linguistic expressions into FCA, these approaches offer more nuanced and accurate representations of complex, real-world scenarios, making them valuable tools for decision-making processes in uncertain and qualitative environments.

Concept lattices, formed by extracting concepts from a formal context via derivation operators, provide a hierarchical structure that captures generalization and specialization relationships among attributes. This hierarchy has made concept lattices a valuable tool for extracting association rules [27,28]. Zou et al. [27,29] developed algorithms to extract association rules based on concept lattices and fuzzy concept lattices, with applications spanning recommendation systems and incremental concept lattice construction. Zhi et al. [30] proposed a method for updating concept lattices in dynamic environments, using minimal generators to optimize computational efficiency in rule extraction. However, these approaches often rely on converting fuzzy information to numeric intervals, which may limit their applicability to qualitative contexts.

To address the aforementioned challenges, this paper presents a novel approach to constructing interval linguistic-valued intuitionistic fuzzy concept lattices and extracting linguistic association rules. While IVIFSs have been effective for quantitative evaluation, their limitations in capturing qualitative evaluation information have necessitated a more flexible framework. The primary contributions of this paper can be summarized as follows:

• Development of interval linguistic-valued intuitionistic fuzzy concept lattices: We introduce a novel framework that extends classical concept lattices by incorporating interval linguistic-valued intuitionistic fuzzy sets. This advancement enables the handling of both quantitative and qualitative uncertainty.
• Enhanced representation of linguistic terms for qualitative analysis: Our approach integrates linguistic terms directly into the lattice structure, allowing for a more accurate representation of the fuzziness and ambiguity inherent in linguistic evaluations. This improvement preserves the original richness of linguistic information without transforming it into numerical values, which often leads to loss of context and precision.
• Efficient extraction of linguistic association rules: We propose a dedicated algorithm to extract association rules from interval linguistic-valued intuitionistic fuzzy concept lattices. This algorithm effectively captures relationships in linguistic data, providing valuable insights for applications where qualitative data interpretation is essential.

The rest of this paper is organized as follows. In Section 2, we propose the interval linguistic-valued intuitionistic fuzzy concept lattice based on the concept lattice and linguistic term set. In Section 3, we generate a linguistic association rule set based on the interval linguistic-valued intuitionistic fuzzy concept lattice; on this basis, we give the corresponding association rule extraction algorithm. Section 4 illustrates the effectiveness of the proposed approach through a bank financial risk assessment example. Section 5 compares our proposed approach with existing association rule extraction approaches. Finally, we conclude the paper with a summary and outlook for further research in Section 6. In the Appendix A, we present some preliminaries.

2. Interval Linguistic-Valued Intuitionistic Fuzzy Concept Lattice

In practical applications in different fields such as evaluation, decision making, and control, different people often have different evaluation values (observed values) of the same object or target from different angles. Many scholars have carried out extended research on intuitionistic fuzzy sets, and introduced the concept of interval-valued intuitionistic fuzzy sets, which can well describe the fuzziness and uncertainty of the “neutral state”. In this section, we propose the interval linguistic-valued intuitionistic fuzzy formal context based on interval-valued intuitionistic fuzzy sets and linguistic term sets. According to the duality principle of concept lattices, we construct the interval linguistic-valued intuitionistic fuzzy concept lattice.

2.1. The Construction of the Interval Linguistic-Valued Intuitionistic Fuzzy Concept Lattice

In this subsection, interval linguistic-valued intuitionistic fuzzy formal contexts and the corresponding concept lattices are introduced.

Definition 1.

Let X be the universe of discourse, then an interval linguistic-valued intuitionistic fuzzy set I on X is defined as follows:

$$I=\left\{\langle x,s_{\mu }\left(x\right),s_{\nu }\left(x\right)\rangle \left|x\in X\right.\right\},$$

(1)

where $s_{\mu }\left(x\right)=[s_{\mu }^{-}\left(x\right),s_{\mu }^{+}\left(x\right)]$ and $s_{\nu }\left(x\right)=[s_{\nu }^{-}\left(x\right),s_{\nu }^{+}\left(x\right)]$ are, respectively, expressed as the interval linguistic-valued membership degree and the interval linguistic-valued non-membership degree of I. For all $x\in X$ on I, $s_0\le s_{\mu }^{+}\left(x\right)+s_{\nu }^{+}\left(x\right)\le s_g$. We say $S_{\pi }\left(x\right)=[s_g-s_{\mu }^{+}\left(x\right)-s_{\nu }^{+}\left(x\right),s_g-s_{\mu }^{-}\left(x\right)-s_{\nu }^{-}\left(x\right)]$ is the interval linguistic-valued intuitionistic fuzzy hesitation degree of I. We denote all interval linguistic-valued intuitionistic fuzzy sets on I as ILV-IFS$\left(I\right)$.

The operation between any two interval-valued intuitionistic fuzzy sets $X_1,X_2\in \mathrm{ILV}-\mathrm{IFS}\left(I\right)$ is defined as follows:

1. 

$X_1^C=\left\{\langle x,[s_{{\nu }_1}^{-}\left(x\right),s_{{\nu }_1}^{+}\left(x\right)],[s_{{\mu }_1}^{-}\left(x\right),s_{{\mu }_1}^{+}\left(x\right)]\rangle \right|x\in X\}$ is the complement of $X_1$;

2. 

$X_1\cap X_2=\langle [s_{{\mu }_1}^{-}\left(x\right)\wedge s_{{\mu }_2}^{-}\left(x\right),s_{{\mu }_1}^{+}\left(x\right)\wedge s_{{\mu }_2}^{+}\left(x\right)],[s_{{\nu }_1}^{-}\left(x\right)\vee s_{{\nu }_2}^{-}\left(x\right),$$s_{{\nu }_1}^{+}\left(x\right)\vee s_{{\nu }_2}^{+}\left(x\right)\left)\right]\rangle$;

3. 

$X_1\cup X_2=\langle [s_{{\mu }_1}^{-}\left(x\right)\vee s_{{\mu }_2}^{-}\left(x\right),s_{{\mu }_1}^{+}\left(x\right)\vee s_{{\mu }_2}^{+}\left(x\right)],[s_{{\nu }_1}^{-}\left(x,a\right)\wedge s_{{\nu }_2}^{-}\left(x\right)),$$s_{{\nu }_1}^{+}\left(x\right)\wedge s_{{\nu }_2}^{+}\left(x\right)\left)\right]\rangle$;

4. 

$X_1\le X_2\iff s_{{\mu }_1}^{-}\left(x\right)\le s_{{\mu }_2}^{-}\left(x\right),s_{{\mu }_1}^{+}\left(x\right)\le s_{{\mu }_2}^{+}\left(x\right),s_{{\nu }_1}^{-}\left(x\right)\ge s_{{\nu }_2}^{-}\left(x\right),s_{{\nu }_1}^{+}\left(x\right)\ge s_{{\nu }_2}^{+}\left(x\right)$.

Definition 2.

An interval linguistic-valued intuitionistic fuzzy formal context is a triple $\left(U,A,S^{*}\right)$, where $U=\left\{x_1,x_2,\cdots ,x_m\right\}$ is a set of objects, $A=\left\{a_1,a_2,\cdots ,a_n\right\}$ is a set of attributes, S is a linguistic term set, and $S^{*}:U\times A\to Int(S\times S)\times Int(S\times S)$ is an interval linguistic-valued intuitionistic fuzzy relationship between U and A such that

$$S^{*}=\left\{\langle x,[s_{\mu }^{-}\left(x,a\right),s_{\mu }^{+}\left(x,a\right)],[s_{\nu }^{-}\left(x,a\right),s_{\nu }^{+}\left(x,a\right)]\rangle \left|\left(x,a\right)\in U\times A\right.\right\}.$$

(2)

Here, $Int(S\times S)$ represents the interval linguistic-valued membership degree. Specifically, $[s_{\mu }^{-}\left(x,a\right),s_{\mu }^{+}\left(x,a\right)]$ represents the interval linguistic-valued membership degree between x and a, $s_{\mu }^{+}\left(x,a\right)$ and $s_{\mu }^{-}\left(x,a\right)$ represent the upper and lower bounds of the interval linguistic-valued membership degree, respectively. $[s_{\nu }^{-}\left(x,a\right),s_{\nu }^{+}\left(x,a\right)]$ represents the interval linguistic-valued non-membership degree between x and a, $s_{\nu }^{+}\left(x,a\right)$ and $s_{\nu }^{-}\left(x,a\right)$ represent the upper and lower bounds of the interval linguistic-valued non-membership degree, respectively. We denote by $S^U$ the set of all interval linguistic-valued intuitionistic fuzzy sets on U.

Linguistic terms within an interval-valued intuitionistic fuzzy framework inherently carry fuzziness and uncertainty, accommodating the ambiguities present in human language. For instance, “high” might be interpreted differently depending on the context, and the interval values enable a range-based approach that expresses this uncertainty.

The incorporation of linguistic terms enhances the interpretability of the concept lattice. Each concept within the concept lattice can be associated with specific linguistic terms, making it easier to interpret the relationships among concepts and understand the data patterns they represent.

Definition 3.

Let $\left(U,A,S^{*}\right)$ be an interval linguistic-valued intuitionistic fuzzy formal context. For any $\ddot{X}\in S^U,B\subseteq A$, two operators “⋉” and “⋊” can be defined as follows:

$${(•)}^{⋉}:S^U\to 2^A,$$

$${\ddot{X}}^{⋉}=\{a\in A|\ddot{X}\left(x\right)\le S^{*}\left(x,a\right),\forall x\in U\},$$

(3)

$${(•)}^{⋊}:2^A\to S^U,$$

$$B^{⋊}=\{\langle [\wedge s_{\mu }^{-}\left(x,a\right),\wedge s_{\mu }^{+}\left(x,a\right)]\rangle ,\langle [\vee s_{\mu }^{-}\left(x,a\right),\vee s_{\mu }^{+}\left(x,a\right)]\rangle |a\in A\}.$$

(4)

Proposition 1.

Suppose $\left(U,A,S^{*}\right)$ is an interval linguistic-valued intuitionistic fuzzy formal context, for any $\ddot{X},{\ddot{X}}_1,{\ddot{X}}_2\in S^U$ and $B,B_1,B_2\subseteq A$, then

1. 

${\ddot{X}}_1\subseteq {\ddot{X}}_2\Rightarrow {\ddot{X}}_2^{⋉}\subseteq {\ddot{X}}_1^{⋉},B_1\subseteq B_2\Rightarrow B_2^{⋊}\subseteq B_1^{⋊}$;

2. 

$\ddot{X}\subseteq {\ddot{X}}^{⋉⋊},B\subseteq B^{⋊⋉}$;

3. 

${({\ddot{X}}_1\cup {\ddot{X}}_2)}^{⋉}={\ddot{X}}_1^{⋉}\cap {\ddot{X}}_2^{⋉},{(B_1\cup B_2)}^{⋊}=B_1^{⋊}\cap B_2^{⋊}$;

4. 

${({\ddot{X}}_1\cap {\ddot{X}}_2)}^{⋉}\supseteq {\ddot{X}}_1^{⋉}\cup {\ddot{X}}_2^{⋉},{(B_1\cap B_2)}^{⋊}\supseteq B_1^{⋊}\cup B_2^{⋊}$;

5. 

${\ddot{X}}^{⋉}={\ddot{X}}^{⋉⋊⋉},B^{⋊}=B^{⋊⋉⋊}$;

6. 

$\ddot{X}\subseteq B^{⋊}\iff B\subseteq {\ddot{X}}^{⋉}$.

Proof. 

According to Definition 3, we can easily prove that properties 1 and 2 hold.

3. According to Definition 3, we have ${({\ddot{X}}_1\cup {\ddot{X}}_2)}^{⋉}={\bigcap }_{x\in {\ddot{X}}_1\cup {\ddot{X}}_2}{x}^{⋉}=\left({\bigcap }_{x\in {\ddot{X}}_1}{x}^{⋉}\right)\cap \left({\bigcap }_{x\in {\ddot{X}}_2}{x}^{⋉}\right)={\ddot{X}}_1^{⋉}\cap {\ddot{X}}_2^{⋉}$. Thus, ${({\ddot{X}}_1\cup {\ddot{X}}_2)}^{⋉}={\ddot{X}}_1^{⋉}\cap {\ddot{X}}_2^{⋉}$ holds. Similarly, we can prove that ${(B_1\cup B_2)}^{⋊}=B_1^{⋊}\cap B_2^{⋊}$.

4. The proof is similar to property 3.

5. According to properties 1 and 2, we have ${\ddot{X}}^{⋉⋊⋉}\subseteq {\ddot{X}}^{⋉}$. Suppose $\ddot{X}={\ddot{X}}^{⋉}$, we have ${\ddot{X}}^{⋉}\subseteq {\ddot{X}}^{⋉⋊⋉}$, i.e., ${\ddot{X}}^{⋉}={\ddot{X}}^{⋉⋊⋉}$. Similarly, we have $B^{⋊}=B^{⋊⋉⋊}$. Therefore, property 3 is proved.

6. “⇒”: Suppose $\ddot{X}\subseteq B^{⋊}$, according to Definition 3 we obtain $B^{⋊⋉}\subseteq {\ddot{X}}^{⋉}$. According to property 2, we have $B\subseteq B^{⋊⋉}$, which yields $B\subseteq {\ddot{X}}^{⋉}$. Therefore, $\ddot{X}\subseteq B^{⋊}\Rightarrow B\subseteq {\ddot{X}}^{⋉}$.

“⇐”: Suppose $B\subseteq {\ddot{X}}^{⋉}$, according to Definition 3 we obtain ${\ddot{X}}^{⋉⋊}\subseteq B^{⋊}$. According to property 2, we have $\ddot{X}\subseteq {\ddot{X}}^{⋉⋊}$, which yields $\ddot{X}\subseteq B^{⋊}$. Therefore, $\ddot{X}\subseteq B^{⋊}\Leftarrow B\subseteq {\ddot{X}}^{⋉}$.

Combining the above arguments, we obtain $\ddot{X}\subseteq B^{⋊}\iff B\subseteq {\ddot{X}}^{⋉}$.    □

Definition 4.

Let $\left(U,A,S^{*}\right)$ be an interval linguistic-valued intuitionistic fuzzy formal context. For any $\ddot{X}\in S^U,B\subseteq A$, if ${\ddot{X}}^{⋉}=B$ and $\ddot{X}=B^{⋊}$, then a pair $(\ddot{X},B)$ is called an interval linguistic-valued intuitionistic fuzzy concept, and $\ddot{X}$ and B are called the extent and intent of the interval linguistic-valued intuitionistic fuzzy concept, respectively.

Definition 5.

Let $\left(U,A,S^{*}\right)$ be an interval linguistic-valued intuitionistic fuzzy formal context, and $\overline{M}IFLL\left(U,A,S^{*}\right)$ be the set of all interval linguistic-valued intuitionistic fuzzy concepts on $\left(U,A,S^{*}\right)$. For any $({\ddot{X}}_1,B_1),({\ddot{X}}_2,B_2)\in \overline{M}IFLL\left(U,A,S^{*}\right)$, the partial-order relationship “≤” between $({\ddot{X}}_1,B_1)$ and $({\ddot{X}}_2,B_2)$ is defined as

$$({\ddot{X}}_1,B_1)\le ({\ddot{X}}_2,B_2)\iff {\ddot{X}}_1\subseteq {\ddot{X}}_2(\iff B_1\supseteq B_2),$$

(5)

where $({\ddot{X}}_1,B_1)$ is called the subconcept of $({\ddot{X}}_2,B_2)$, and $({\ddot{X}}_2,B_2)$ is called the superconcept of $({\ddot{X}}_1,B_1)$. If there is no $({\ddot{X}}_3,B_3)$ that satisfies $({\ddot{X}}_1,B_1)\le ({\ddot{X}}_3,B_3)\le ({\ddot{X}}_2,B_2)$, then $({\ddot{X}}_1,B_1)$ is a child concept of $({\ddot{X}}_2,B_2)$, and $({\ddot{X}}_2,B_2)$ is the parent concept of $({\ddot{X}}_1,B_1)$, which is recorded as $({\ddot{X}}_1,B_1)\prec ({\ddot{X}}_2,B_2)$.

Theorem 1.

Let $\left(U,A,S^{*}\right)$ be an interval linguistic-valued intuitionistic fuzzy formal context, and $\overline{M}IFLL\left(U,A,S^{*}\right)$ be the set of all interval linguistic-valued intuitionistic fuzzy concepts on $\left(U,A,S^{*}\right)$. For any $({\ddot{X}}_1,B_1),({\ddot{X}}_2,B_2)\in \overline{M}IFLL\left(U,A,S^{*}\right)$, then

$$({\ddot{X}}_1,B_1)\wedge ({\ddot{X}}_2,B_2)=({\ddot{X}}_1\cap {\ddot{X}}_2,{(B_1\cup B_2)}^{⋊⋉}),$$

(6)

$$({\ddot{X}}_1,B_1)\vee ({\ddot{X}}_2,B_2)=({({\ddot{X}}_1\cup {\ddot{X}}_2)}^{⋉⋊},B_1\cap B_2),$$

(7)

are also interval linguistic-valued intuitionistic fuzzy concepts.

Proof. 

According to 3, 4, and 5 of Proposition 1, we have ${(B_1\cup B_2)}^{⋊}={(B_1\cup B_2)}^{⋊⋉⋊}=B_1^{⋊}\cap B_2^{⋊}={\ddot{X}}_1\cap {\ddot{X}}_2$ and ${({\ddot{X}}_1\cap {\ddot{X}}_2)}^{⋉}={(B_1\cup B_2)}^{⋉⋊}$. According to Definition 3, we can prove that $({\ddot{X}}_1\cap {\ddot{X}}_2,{(B_1\cup B_2)}^{⋊⋉})\in \overline{M}IFLL\left(U,A,S^{*}\right)$. Similarly, $({({\ddot{X}}_1\cup {\ddot{X}}_2)}^{⋉⋊},B_1\cap B_2)\in \overline{M}IFLL\left(U,A,S^{*}\right)$.    □

$(\overline{M}IFLL\left(U,A,S^{*}\right),\le )$ is a complete lattice, called an interval linguistic-valued intuitionistic fuzzy concept lattice, denoted as $\overline{M}IFLL\left(U,A,S^{*}\right)$.

Theorem 2.

Let $\left(U,A,S^{*}\right)$ be an interval linguistic-valued intuitionistic fuzzy formal context, and $\overline{M}IFLL\left(U,A,S^{*}\right)$ be the set of all interval linguistic-valued intuitionistic fuzzy concepts on $\left(U,A,S^{*}\right)$. For any $({\ddot{X}}_1,B_1),({\ddot{X}}_2,B_2)\in \overline{M}IFLL\left(U,A,S^{*}\right)$, then $({\ddot{X}}_1^{⋉⋊},{\ddot{X}}_1^{⋉}),(B_1^{⋊},B_1^{⋊⋉})\in \overline{M}IFLL\left(U,A,S^{*}\right)$.

Proof. 

According to Proposition 1 and Definition 3, we can easily obtain $({\ddot{X}}_1^{⋉⋊},{\ddot{X}}_1^{⋉}),(B_1^{⋊},B_1^{⋊⋉})\in \overline{M}IFLL\left(U,A,S^{*}\right)$.    □

The interval linguistic-valued intuitionistic fuzzy concept lattice offers several significant advantages over the classical concept lattice, especially in scenarios where data are uncertain, imprecise, or qualitative in nature. Here are some key benefits:

• The interval linguistic-valued intuitionistic fuzzy concept lattice incorporates both fuzziness and intuitionistic uncertainty through interval values. This allows it to represent the degree of membership, non-membership, and hesitation more effectively than classical concept lattices, which are limited to binary or crisp relationships.
• Unlike classical concept lattices, which rely on binary or numeric values, the interval linguistic-valued intuitionistic fuzzy concept lattice can directly handle linguistic terms (e.g., “high”, “medium”, “low”). This reduces the need for complex transformations and preserves the richness of linguistic information, reducing information loss.
• This lattice structure captures both positive and negative relationships, allowing for a more nuanced representation of concepts and their connections. It can express both supportive and contrary relationships between objects and attributes, providing a fuller understanding of the data.
• With its ability to incorporate both uncertainty and linguistic information, the interval linguistic-valued intuitionistic fuzzy concept lattice supports more informed and context-sensitive decision making. It provides a more accurate basis for rule extraction and association mining, yielding insights that would be difficult to capture using a classical concept lattice.

2.2. Algorithm Description

From Definitions 3 and 4, all the interval linguistic-valued intuitionistic fuzzy concepts are obtained. According to Definition 5, we can obtain the partial order of all interval linguistic-valued intuitionistic fuzzy concepts to construct the interval linguistic-valued intuitionistic fuzzy concept lattice. This subsection provides a description of the algorithm for generating the interval linguistic-valued intuitionistic fuzzy concept lattice.

Through the analysis in the previous section, we give the Interval Linguistic-Valued Intuitionistic Fuzzy Concept Lattice Generation (IVI-FCLG) algorithm (see Algorithm 1).

In the following, we analyze the time complexity of the IVI-FCLG algorithm.

In Algorithm 1, we assume that the interval linguistic-valued intuitionistic fuzzy formal context $\left(U,A,S^{*}\right)$ has o attributes and p objects. According to steps 1–2, the objects and attributes needs to be traversed when computing $\ddot{X}$ and B. According to steps 3–14, when computing the parent–child concepts, it is necessary to determine ${\ddot{X}}_i\subseteq {\ddot{X}}_j$ or ${\ddot{X}}_j\subseteq {\ddot{X}}_i$. Therefore, the overall time complexity of the IVI-FCLG is $O\left(p{o}^2\right)$.

Algorithm 1: IVI-FCLG

Input: Interval linguistic-valued intuitionistic fuzzy formal context $\left(U,A,S^{*}\right)$  Output: Interval linguistic-valued intuitionistic fuzzy concept lattice $\overline{M}IFLL\left(U,A,S^{*}\right)$

IVI-FCLG serves as a preprocessing step for linguistic association rule extraction. It prepares the foundational structure (i.e., the interval linguistic-valued intuitionistic fuzzy concept lattice) that is essential for linguistic association rule extraction.

3. Linguistic Association Rule Extraction Based on Interval Linguistic-Valued Intuitionistic Fuzzy Concept Lattice

The Hasse diagram can display the partial order in the concept lattice concisely, which is conducive to the rapid mining of effective association rules. In order to reduce the loss of information generated by the concept lattice when generating association rules, this section extracts linguistic association rules from the two perspectives of membership and non-membership of linguistic expressions through the interval linguistic-valued intuitionistic fuzzy set.

Definition 6.

Let $\left(U,A,S^{*}\right)$ be an interval linguistic-valued intuitionistic fuzzy formal context, and $\overline{M}IFLL\left(U,A,S^{*}\right)$ be the interval linguistic-valued intuitionistic fuzzy concept lattice corresponding to $\left(U,A,S^{*}\right)$. For any $(\ddot{X},B)\in \overline{M}IFLL\left(U,A,S^{*}\right)$, the expectation $E\left(B\right)$ of concept extent to intent is defined as

$$E\left(B\right)={\displaystyle \frac{1}{2\left|\ddot{X}\right|}}\sum _{j\in \ddot{X}}\left({\displaystyle \frac{1}{2}}\left[s_{\mu }^{-}\left(x_j,a\right)+s_{\mu }^{+}\left(x_j,a\right)\right]+\left[s_g-{\displaystyle \frac{1}{2}}\left[s_{\nu }^{-}\left(x_j,a\right)+s_{\nu }^{+}\left(x_j,a\right)\right]\right]\right)$$

(8)

The expectation value of a concept can be thought of as a measure of its linguistic membership strength within the dataset, which reflects how frequently a particular combination of attributes appears or holds significant meaning across the dataset.

In real-world datasets, some concepts might have very low membership values, indicating they are either rare or hold minimal significance in the context of the application. By setting a threshold $\theta$ for $E\left(B\right)$, we can effectively filter out these less relevant nodes, ensuring that only nodes with substantial relevance (i.e., those meeting or exceeding $\theta$) are considered in the rule generation process.

When $E\left(B\right)$ is greater than a certain linguistic membership threshold $\theta$, the node is called a frequent node. This helps filter out concepts that appear infrequently or lack strong linguistic association, thereby focusing the rule extraction process on nodes that represent commonly observed patterns.

Proposition 2.

Suppose $\left(U,A,S^{*}\right)$ is an interval linguistic-valued intuitionistic fuzzy formal context, and $\overline{M}IFLL\left(U,A,S^{*}\right)$ is the interval linguistic-valued intuitionistic fuzzy concept lattice corresponding to $\left(U,A,S^{*}\right)$. For any $(\ddot{X},B)\in \overline{M}IFLL\left(U,A,S^{*}\right)$, the expectation $E\left(B\right)$ satisfies

$$0<E\left(B\right)\le 1.$$

(9)

Proof. 

It follows immediately from Definition 6.    □

In the interval linguistic-valued intuitionistic fuzzy concept lattice, for a pair of frequent parent–child nodes $(C_i,C_j)(C_i=({\ddot{X}}_1,B_1),C_j=({\ddot{X}}_2,B_2))$, we call $B_1\Rightarrow B_2-B_1$ a linguistic association rule.

Definition 7.

Let $R:B_1\Rightarrow B_2-B_1$ be a linguistic association rule, then the confidence and support of R are defined as

$$Conf\left(R\right)={\displaystyle \frac{min\left(E\left(B_1\right),E\left(B_2\right)\right)}{max\left(E\left(B_1\right),E\left(B_2\right)\right)}},$$

(10)

$$\begin{array}{cc}\hfill Sup\left(R\right)& =E\left(B_1\right)={\displaystyle \frac{1}{2\left|{\ddot{X}}_1\right|}}\sum _{j\in {\ddot{X}}_1}({\displaystyle \frac{1}{2}}\left[s_{\mu }^{-}\left(x_j,a\right)+s_{\mu }^{+}\left(x_j,a\right)\right]+\hfill \\ \hfill & \left[s_g-{\displaystyle \frac{1}{2}}\left[s_{\nu }^{-}\left(x_j,a\right)+s_{\nu }^{+}\left(x_j,a\right)\right]\right]).\hfill \end{array}$$

(11)

$Sup\left(R\right)$ reflects the frequency of simultaneous occurrence of $B_1$ and $B_2-B_1$ in the attribute set. $Conf\left(R\right)$ reflects the probability of $B_2-B_1$ occurring when $B_1$ occurs.

Remark 1.

 1. The support $Sup\left(R\right)$ represents the frequency at which both the antecedent $B_1$ and the consequent $B_2-B_1$ of the association rule occur together among all attributes. A higher support value indicates a stronger connection between $B_1$ and $B_2-B_1$, suggesting a more frequent co-occurrence of these elements.

2. 

The confidence $Conf\left(R\right)$ reflects the probability that the consequent $B_2-B_1$ of the association rule appears when the antecedent $B_1$ is present. A higher confidence value indicates a more reliable association between the antecedent and the consequent in the linguistic association rule extraction.

Through the above discussion, we give a Linguistic Association Rule Extraction (LARE) algorithm based on an interval linguistic-valued intuitionistic fuzzy concept lattice (see Algorithm 2).

Algorithm 2: LARE

In the following, we analyze the time complexity of LARE. Since the input to Algorithm 2 is an interval linguistic-valued intuitionistic fuzzy concept lattice, we do not consider the time complexity of constructing the interval linguistic-valued intuitionistic fuzzy concept lattice.

In Algorithm 2, we assume that the interval linguistic-valued intuitionistic fuzzy concept lattice $\overline{M}IFLL\left(U,A,S^{*}\right)$ has q concepts. According to steps 1–9, the time complexity of computing the expectation $E\left(B_i\right)$ for each concept is $O\left(q\right)$. According to steps 11–19, assume that $r(r<q)$ frequent nodes are generated from steps 1–9, and the time complexity of computing the confidence of two frequent nodes is $O\left(r^2\right)$. Therefore, the overall time complexity of the LARE is $O(q+r^2)$.

Several bottlenecks in the current design could impact the performance of the LARE algorithm, especially with larger datasets:

• Exponential growth in subset calculation: Generating subsets for attributes ($2^{\left|A\right|}$) poses a major challenge when $\left|A\right|$ is large, as the number of subsets increases exponentially. This step is especially challenging in applications with a high-dimensional attribute space, as it quickly becomes computationally prohibitive.
• High complexity of pairwise node comparison: Comparing each concept to others in terms of confidence relationships within the concept lattice (${\left|q\right|}^2$) can lead to significant computational overhead, particularly in dense lattices with many hierarchical connections.
• Hierarchical relationships and Hasse diagram construction: Building and traversing the Hasse diagram for lattice relationships, particularly for rule extraction in complex lattices, may also introduce processing delays, especially when the lattice structure is highly interconnected.

LARE is a subsequent step that works exclusively on the output of IVI-FCLG. It uses the lattice structure as input to mine linguistic association rules.

The Hasse diagram serves as a crucial visualization tool for illustrating the partial order of concepts in an interval linguistic-valued intuitionistic fuzzy concept lattice. This structured visualization is particularly beneficial for rule extraction in the following ways:

• The hierarchical nature of the Hasse diagram supports parent–child relationships, which are key to mining association rules. In LARE, for instance, rules are generated by examining these relationships to identify how attributes transition from one concept to another along the hierarchy. This approach allows for the systematic extraction of rules that reflect natural, progressive relationships, such as how adding an attribute might lead to a more specific concept.
• By navigating the Hasse diagram, users can efficiently calculate metrics such as membership values and confidence for rules. Since each connection in the diagram represents a specific conceptual linkage, it becomes easier to apply thresholds (e.g., confidence and support) to filter and select only those rules that meet desired criteria. This filtering process ensures that the extracted rules are both relevant and interpretable.
• The Hasse diagram’s visual structure helps users intuitively understand the logic behind each rule. By observing the transitions between concepts, users can interpret rules in terms of natural language expressions. For instance, if a concept associated with “high risk” in financial assessments is connected to another concept with “very high risk”, the rule extracted along this path can be directly interpreted as “if high risk, then very high risk”.

4. Case Study

In this section, we utilize our proposed IVI-FCLG and LARE approaches to address the problem of bank financial risk assessment in a linguistic environment. This paper focuses on the construction of an interval linguistic-valued intuitionistic fuzzy concept lattice and linguistic association rule extraction.

4.1. Background Description

In the banking sector, financial risk assessment is a critical process used to evaluate a customer’s likelihood of default, financial stability, and overall creditworthiness. Traditional risk assessment relies on quantitative metrics, such as credit scores, debt-to-income ratios, and historical loan repayment data. However, a purely numerical approach may overlook nuanced insights, particularly when data are uncertain or subjective. In such cases, linguistic assessments provide an alternative that captures qualitative aspects of financial risk using descriptors such as “high risk”, “moderate stability”, or “low reliability”. Linguistic association rule extraction is used to identify associations in this qualitative data to support decision making for loans, credit lines, and investments.

4.2. Problem Solution

We consider an interval linguistic-valued intuitionistic fuzzy formal context $\left(U,A,S^{*}\right)$, as shown in

Table 1. Interval linguistic-valued intuitionistic fuzzy formal context $\left(U,A,S^{*}\right)$.

U	a	b	c	d
$x_1$	$〈\left[s_1,s_3\right],\left[s_4,s_5\right]〉$	$〈\left[s_4,s_6\right],\left[s_1,s_2\right]〉$	$〈\left[s_4,s_5\right],\left[s_0,s_3\right]〉$	$〈\left[s_5,s_6\right],\left[s_1,s_2\right]〉$
$x_2$	$〈\left[s_3,s_4\right],\left[s_1,s_3\right]〉$	$〈\left[s_4,s_6\right],\left[s_0,s_2\right]〉$	$〈\left[s_4,s_4\right],\left[s_2,s_3\right]〉$	$〈\left[s_5,s_7\right],\left[s_0,s_1\right]〉$
$x_3$	$〈\left[s_6,s_7\right],\left[s_0,s_1\right]〉$	$〈\left[s_3,s_4\right],\left[s_1,s_2\right]〉$	$〈\left[s_4,s_5\right],\left[s_0,s_3\right]〉$	$〈\left[s_4,s_6\right],\left[s_0,s_2\right]〉$
$x_4$	$〈\left[s_4,s_6\right],\left[s_1,s_2\right]〉$	$〈\left[s_2,s_4\right],\left[s_3,s_4\right]〉$	$〈\left[s_2,s_3\right],\left[s_4,s_5\right]〉$	$〈\left[s_5,s_7\right],\left[s_0,s_1\right]〉$
$x_5$	$〈\left[s_4,s_5\right],\left[s_0,s_3\right]〉$	$〈\left[s_4,s_6\right],\left[s_1,s_2\right]〉$	$〈\left[s_4,s_5\right],\left[s_2,s_3\right]〉$	$〈\left[s_5,s_6\right],\left[s_0,s_1\right]〉$

, where $U=\left\{x_1,x_2,x_3,x_4,x_5\right\}$ represents five experts, $A=\left\{a,b,c,d\right\}$ represents the set of qualitative risk assessment indices, and $S=\{s_1,s_2,s_3,s_4,s_5,s_6,s_7,s_8\}$ represents the experts’ linguistic assessment values for bank financial risk.

According to Algorithm 1, we can obtain all interval linguistic-valued intuitionistic fuzzy concepts on the interval linguistic-valued intuitionistic fuzzy formal context $\left(U,A,S^{*}\right)$, as shown in

Table 2. All interval linguistic-valued intuitionistic fuzzy concepts.

Index	Extent	Intent
$0\#$	$\{〈\left[s_5,s_6\right],\left[s_0,s_2\right]〉,〈\left[s_5,s_7\right],\left[s_0,s_1\right]〉,〈\left[s_6,s_7\right],\left[s_0,s_1\right]〉,〈\left[s_5,s_7\right],\left[s_0,s_1\right]〉,〈\left[s_5,s_6\right],\left[s_0,s_1\right]〉\}$	∅
$1\#$	$\{〈\left[s_1,s_3\right],\left[s_4,s_5\right]〉,〈\left[s_3,s_4\right],\left[s_1,s_3\right]〉,〈\left[s_6,s_7\right],\left[s_0,s_1\right]〉,〈\left[s_4,s_6\right],\left[s_1,s_2\right]〉,〈\left[s_4,s_5\right],\left[s_0,s_3\right]〉\}$	$\left\{a\right\}$
$2\#$	$\{〈\left[s_4,s_5\right],\left[s_0,s_3\right]〉,〈\left[s_4,s_4\right],\left[s_2,s_3\right]〉,〈\left[s_4,s_5\right],\left[s_0,s_3\right]〉,〈\left[s_2,s_3\right],\left[s_4,s_5\right]〉,〈\left[s_4,s_5\right],\left[s_2,s_3\right]〉\}$	$\left\{c\right\}$
$3\#$	$\{〈\left[s_5,s_6\right],\left[s_0,s_2\right]〉,〈\left[s_5,s_7\right],\left[s_0,s_1\right]〉,〈\left[s_4,s_6\right],\left[s_0,s_2\right]〉,〈\left[s_5,s_7\right],\left[s_0,s_1\right]〉,〈\left[s_5,s_6\right],\left[s_0,s_1\right]〉\}$	$\left\{d\right\}$
$4\#$	$\{〈\left[s_4,s_6\right],\left[s_1,s_2\right]〉,〈\left[s_4,s_6\right],\left[s_0,s_2\right]〉,〈\left[s_3,s_4\right],\left[s_1,s_2\right]〉,〈\left[s_2,s_4\right],\left[s_3,s_4\right]〉,〈\left[s_4,s_6\right],\left[s_1,s_2\right]〉\}$	$\{b,d\}$
$5\#$	$\{〈\left[s_4,s_5\right],\left[s_1,s_3\right]〉,〈\left[s_4,s_4\right],\left[s_2,s_3\right]〉,〈\left[s_4,s_5\right],\left[s_0,s_3\right]〉,〈\left[s_2,s_3\right],\left[s_4,s_5\right]〉,〈\left[s_4,s_5\right],\left[s_2,s_3\right]〉\}$	$\{c,d\}$
$6\#$	$\{〈\left[s_1,s_3\right],\left[s_4,s_5\right]〉,〈\left[s_3,s_4\right],\left[s_1,s_3\right]〉,〈\left[s_3,s_4\right],\left[s_1,s_2\right]〉,〈\left[s_2,s_4\right],\left[s_3,s_4\right]〉,〈\left[s_4,s_5\right],\left[s_1,s_3\right]〉\}$	$\{a,b,d\}$
$7\#$	$\{〈\left[s_4,s_5\right],\left[s_1,s_3\right]〉,〈\left[s_4,s_4\right],\left[s_2,s_3\right]〉,〈\left[s_3,s_4\right],\left[s_1,s_3\right]〉,〈\left[s_2,s_3\right],\left[s_4,s_5\right]〉,〈\left[s_4,s_5\right],\left[s_2,s_3\right]〉\}$	$\{b,c,d\}$
$8\#$	$\{〈\left[s_1,s_3\right],\left[s_4,s_5\right]〉,〈\left[s_3,s_4\right],\left[s_2,s_3\right]〉,〈\left[s_4,s_5\right],\left[s_0,s_3\right]〉,〈\left[s_2,s_3\right],\left[s_4,s_5\right]〉,〈\left[s_4,s_5\right],\left[s_2,s_3\right]〉\}$	$\{a,c,d\}$
$9\#$	$\{〈\left[s_1,s_3\right],\left[s_4,s_5\right]〉,〈\left[s_3,s_4\right],\left[s_2,s_3\right]〉,〈\left[s_3,s_4\right],\left[s_1,s_3\right]〉,〈\left[s_2,s_3\right],\left[s_4,s_5\right]〉,〈\left[s_4,s_5\right],\left[s_2,s_3\right]〉\}$	$\{a,b,c,d\}$

.

In Algorithm 1, through steps 3–14, we can determine the partial order between interval linguistic-valued intuitionistic fuzzy concepts and obtain the interval linguistic-valued intuitionistic fuzzy concept lattice $\overline{M}IFLL\left(U,A,S^{*}\right)$, as shown in Figure 1.

According to Algorithm 1, we set the average linguistic membership threshold $\theta =s_{4.5}$, the confidence threshold $\varphi =0.83$, and the linguistic association rule set $R_s$ extracted from Figure 1 is shown in

Table 3. The linguistic association rule set $R_s$.

Index	${\mathit{R}}_{\mathit{s}}$	$\mathit{E}\left({\mathit{B}}_1\right)$	$\mathit{E}\left({\mathit{B}}_2\right)$	$\mathbf{Conf}\left(\mathit{R}\right)$
0	$\{c,d\}\Rightarrow \left\{a\right\}$	$s_{4.60}$	$s_{4.15}$	0.90
1	$\{c,d\}\Rightarrow \left\{b\right\}$	$s_{4.65}$	$s_{4.60}$	0.99
2	$\{b,d\}\Rightarrow \left\{c\right\}$	$s_{5.20}$	$s_{4.65}$	0.89
3	$\{b,d\}\Rightarrow \left\{a\right\}$	$s_{5.20}$	$s_{4.30}$	0.83
4	$\left\{c\right\}\Rightarrow \left\{d\right\}$	$s_{4.75}$	$s_{4.60}$	0.97

.

Through the above analysis, we find that LARE describes the interval linguistic-valued intuitionistic fuzzy relationships between objects and attributes through the interval linguistic-valued intuitionistic fuzzy concept lattice. The linguistic association rules can be extracted through the intent of the parent and child concept nodes of the concept lattice to generate user understandable interest and obtain user feedback on the assessment logic.

5. Comparative Analysis and Discussion

In this section, we provide a comparative analysis of our proposed approach, LARE, with several other association rule extraction approaches based on key dimensions such as starting point, rule type, background knowledge, and downstream task applicability.

We compare our approach, LARE, with several other association rule extraction approaches, including the Association Rule Mining approach based on Fuzzy Concept Lattice (ARM-FCL) [27], the Fuzzy Association Rule Mining approach based on Hedge Algebras (FARM-HA) [31], the Obtain Positive and Negative Classification Rules approach (OPNCR) [32], the Association Rule Mining approach based on Concept Lattice (ARM-CL) [29], and the Association Rules Generation with Designated Consequence approach (ARGDC) [33]. A comparison of the results of these approaches is presented in

Table 4. Comparison of different approaches.

Approach	Starting Point 1	Rule Type 2	Background Knowledge 3	Downstream Task 4
ARM-FCL	Fuzzy concept lattice	Fuzzy values	No	None
FARM-HA	Transactional database, qualitative evaluation table	Linguistic values	Yes	None
OPNCR	Concept lattice	Boolean values	No	Text mining
ARM-CL	Concept lattice	Boolean values	No	Recommendation system
ARGDC	Concept lattice	Boolean values	No	Materials research
LARE	Interval linguistic-valued intuitionistic fuzzy concept lattice	Linguistic values	Yes	None

¹ Starting point: The initial data structure or framework used. ² Rule type: The type of rules generated or used in the approach. ³ Background knowledge: Indicates whether the approach leverages background knowledge. ⁴ Downstream task: The specific application or domain where the approach is applied.

.

According to Table 4, we compare our proposed approach with other association rule extraction approaches across four key dimensions:

• Starting point: Among the various association rule extraction approaches, LARE is unique in being based on the interval linguistic-valued intuitionistic fuzzy concept lattice. This structure allows LARE to explicitly handle the inherent uncertainty in linguistic values while analyzing the relationships between users and items from both positive and negative perspectives. In contrast, three other approaches—OPNCR, ARM-CL, and ARGDC—derive association rules using concept lattices, which focus on capturing the hierarchical relationships among concepts. FARM-HA, on the other hand, extracts association rules from a transaction database combined with a qualitative evaluation table. Unlike transaction databases, concept lattices offer a way to visualize the generalization and specialization relationships between concepts, making it easier to pinpoint relevant nodes for association rule extraction. Notably, ARM-FCL, based on fuzzy concept lattices, accounts for the fuzzy relationships between users and items, further emphasizing the adaptability of concept lattices in modeling complex relationships.
• Rule type: Several approaches (OPNCR, ARM-CL, ARGDC) rely on Boolean values for association rule extraction. While effective for crisp, binary relationships, Boolean-based approaches have limitations in capturing the uncertain and graded relationships between users and items. ARM-FCL introduces fuzzy values to model these relationships, providing a more nuanced understanding of the degrees of association. However, none of these approaches can process qualitative linguistic-valued data. Both FARM-HA and LARE overcome this limitation by directly representing linguistic-valued data: FARM-HA utilizes qualitative evaluation tables, while LARE employs interval linguistic-valued intuitionistic fuzzy sets. In this way, these approaches reduce information loss caused by the transformation of linguistic-valued data into numerical formats, preserving the richness of the original linguistic information.
• Background knowledge: Four of the approaches do not incorporate background knowledge into the rule extraction process. FARM-HA stands out by using a qualitative evaluation table as background knowledge, guiding association rule extraction in transaction databases through hedge algebras. In comparison, LARE leverages the semantic ordering of linguistic terms as background knowledge. This allows LARE to guide the extraction of association rules within the interval linguistic-valued intuitionistic fuzzy concept lattice, incorporating both positive and negative linguistic information. This dual-sided analysis provides a more comprehensive understanding of the relationships embedded in the data, making LARE more versatile than approaches like FARM-HA that rely on a single perspective.
• Downstream task: While ARM-FCL, FARM-HA, and LARE are not directly applied to specific downstream tasks, approaches like ARM-CL, OPNCR, and ARGDC, which are based on concept lattices, have been successfully applied to practical areas such as recommender systems, text mining, and materials research:
  • ARM-CL leverages association rule mining within concept lattices to improve recommendation accuracy. For instance, in e-commerce platforms, ARM-CL can analyze users’ purchase histories to generate association rules that suggest potential items of interest. If a user has purchased items A and B, the system may recommend item C based on discovered rules such as $\{A,B\}\to C$, derived from the concept lattice.
  • OPNCR is effectively used in text mining to uncover patterns and topics within textual data. For example, in news article classification, OPNCR can generate a concept lattice based on the logical relationships between words and sentences. Articles containing keywords like “economy”, “growth”, and “stocks” could be automatically categorized under the “finance” section, helping streamline large-scale text processing.
  • ARGDC has shown promise in materials science for exploring correlations between material properties and performance. In material design, concept lattices constructed by ARGDC help identify relationships between attributes such as density and thermal conductivity, and performance metrics like hardness and electrical conductivity. By analyzing historical material data, ARGDC can suggest new material combinations that meet specific performance criteria, accelerating the discovery of novel materials.
  This highlights the flexibility and utility of concept lattice-based approaches for association rule extraction, particularly in domains where visualization, interpretability, and portability are crucial. The ability to translate extracted rules into actionable insights demonstrates the practical advantages of using concept lattices in real-world applications, further underscoring the potential for LARE in future downstream tasks.

In summary, while traditional methods like ARM-FCL and FARM-HA have contributed to the advancement of association rule mining in fuzzy and hybrid contexts, they exhibit certain limitations when handling complex, qualitative information. These limitations include a reliance on fixed numerical thresholds, insufficient flexibility in processing linguistic uncertainty, and suboptimal performance in highly dynamic environments.

In contrast, the proposed LARE algorithm leverages interval linguistic-valued intuitionistic fuzzy sets to overcome these challenges. By integrating the concept lattice framework with linguistic membership and confidence thresholds, LARE excels in the following:

• Accurately capturing qualitative nuances inherent in human decision making.
• Generating more interpretable and meaningful rules that align with real-world linguistic descriptions.
• Enhancing computational efficiency by reducing redundant rules and focusing on high-confidence implications.

Future research could explore applying LARE to specific downstream tasks like recommender systems, building on its dual-sided analysis of linguistic information.

6. Conclusions

In this paper, we propose a novel construction algorithm for the interval linguistic-valued intuitionistic fuzzy concept lattice, building on the foundations of interval intuitionistic fuzzy sets and FCA. We also explore the key properties of the interval linguistic-valued intuitionistic fuzzy concept lattice, providing a theoretical basis for its application. On the basis of this lattice, we further introduce a linguistic association rule extraction algorithm. The interval linguistic-valued intuitionistic fuzzy concept lattice captures linguistic information from both positive and negative perspectives, thereby reducing information loss commonly encountered in other approaches.

By leveraging the partial order relationships between interval linguistic-valued intuitionistic fuzzy concepts, the proposed algorithm is capable of extracting higher-quality linguistic association rules. These rules are not only more nuanced but also reflect the complex interrelationships between concepts, enhancing the overall effectiveness of association rule mining in environments characterized by imprecision and vagueness. This approach offers significant advantages in terms of capturing the full spectrum of linguistic information, improving both the interpretability and practical applicability of the mined association rules.

With the increasing volume of data, it is becoming essential to consider the reduction approach for the interval linguistic-valued intuitionistic fuzzy concept lattice to ensure its practical application in real-world scenarios. This reduction technique would aim to minimize the generation of redundant linguistic association rules, enhancing the efficiency and relevance of the rules derived from the lattice. Furthermore, compared to concept lattices, the interval linguistic-valued intuitionistic fuzzy concept lattice is significantly larger in scale, leading to increased computational complexity. Addressing this issue requires the development of effective pruning techniques to simplify the structure of the lattice, reducing its complexity while maintaining the integrity of the linguistic information. Future research should focus on exploring these reduction and pruning strategies to make the interval linguistic-valued intuitionistic fuzzy concept lattice more scalable and applicable in large-scale data environments. These improvements would not only streamline the construction process but also enhance the practicality of the lattice in applications such as data mining, recommendation systems, and decision-making processes.