Abstract
Knowledge reduction is a crucial research topic in formal concept analysis. Given the ability of threeway concept analysis to capture attribute information in a more comprehensive and detailed manner, this paper focuses on investigating object reduction methods and their applications based on threeway concept lattice. We achieve this by integrating necessity and possibility operators in a formal context containing linguistic data. Our proposed approach involves constructing an attributeinduced threeway objectoriented linguistic concept lattice to classify the object set into three regions based on threeway decision. Additionally, we introduce an approach of attributeinduced threeway objectoriented linguistic granular reduction, which employs granular concepts, granular discernibility functions, and granular consistent sets to enhance the efficiency of extracting uncertainty information. We obtain the relationships between the five consistent sets of attributeinduced threeway objectoriented linguistic concepts and further analyze the corresponding object reduction relations. To demonstrate the feasibility and validity of our proposed method, we apply it to address optimization decisionmaking challenges in corporate projects using different attributeinduced threeway objectoriented linguistic concepts.
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1 Introduction
Formal concept analysis (FCA) is a powerful method for data analysis and knowledge processing, initially proposed by Wille [1]. FCA utilizes formal contexts to describe binary relationships between objects and attributes. At its core, FCA revolves around the formal concept, which consists of two vital components: extent and intent. These components intricately determine each other, serving as the foundation for all formal concepts. The collection of formal concepts can be visualized as a hierarchical structure called a concept lattice, enabling a clear representation of knowledge in FCA’s representation theory. Over the years, researchers have made significant advancements in various research areas based on this theory, including concept lattice reduction, rule extraction, granular description, and concept cognitive learning [2,3,4,5,6,7].
Threeway decision theory is a practical approach that can be used to make decisions based on three possible outcomes: positive, negative, and uncertain. This approach is particularly useful in situations where there is incomplete or ambiguous information. In light of this, the threeway concept lattice [8] integrates the idea of threeway decisions into formal concept analysis, providing a more comprehensive understanding of decisionmaking problems. The threeway concept lattice not only reflects the specific expression and significance of FCA but also embodies the idea of three divisions in threeway decisions, which recognizes the importance of uncertainty in decisionmaking and provides a framework for evaluating and managing it. By utilizing the threeway concept lattice, decisionmakers can evaluate multiple alternatives and their potential outcomes, facilitating better communication and collaboration [9,10,11,12]. For example, Qian et al. [13] have established a threeway concept lattice method by constructing a new isomorphic formal context that combines the given formal context with its complementary context. The objectinduced threeway operators and attributeinduced threeway operators have been introduced in [14], and the corresponding candidate and redundant concepts have been presented to build the threeway concept lattice simply and effectively. Considering that the threeway concept lattice can express both positive and negative information in a formal context, a rule extraction method of the threeway concept lattice has been discussed [15].
Besides, various concepts have been proposed to describe different objective things, such as fuzzy concepts [16], attributeoriented concepts [17], objectoriented concepts [18], approximate concepts [19] and so on. Among them, the objectoriented concept lattice and the attributeoriented concept lattice have also attracted widespread attention, as they can provide new information from another complementary perspective that cannot be expressed by operators of classical formal concepts. In addition, granular computing aims to transform complex data sets into several blocks according to their characteristics and performance, so as to establish an effective largescale data set computing model. Therefore, the combination of object (attribute)oriented concept lattice and granular computing is more meaningful [20, 21]. Inspired by this view, Shao et al. [22] have proposed two zoom algorithms to complete fast construction of the attribute (object)oriented multigranularity concept lattices. From the perspective of possibility operators and necessity operators, Li et al. [23] have discussed techniques such as granular description and cognitive concept learning in an incomplete formal context, which greatly improved the time complexity of mining decision rules. Combined with the idea of threeway decisions, the relations between objectoriented concept lattice, attributeoriented concept lattice, threeway objectoriented concept lattice, and threeway attributeoriented concept lattice have been studied, and a new construction approach based on apposition and subposition have been proposed [24]. To solve special practical problems such as import and export trade, Zhi et al. [25] have introduced a new threeway dual concept lattice and analyzed four models based on threeway concept analysis by six inductive operators.
Knowledge reduction is a key issue in FCA and rough set theory [26,27,28,29], which aims at making the data in the database more concise and saving storage space. In terms of FCA, knowledge reduction can be roughly divided into two parts, one is the reduction in the formal context, and the other is the reduction in the formal decision context. These reduction methods are scattered in the categories of concept reduction [30,31,32], attribute reduction [33,34,35] and object reduction [36, 37], among which attribute reduction is one of the most commonly used methods for finding the minimal subset of attributes that can preserve an important condition, such as lattice structure, granule information and meet(join)irreducible elements, and similar methods also can be applied to object reduction. From the perspective of formal context, Ren et al. [38] have studied four reduction methods on attributeinduced threeway concept lattice and objectinduced threeway concept lattice by discernibility attribute set. Aiming at the complex derivation operator in the fuzzy formal context, Shao et al. [39] have proposed a knowledge reduction method of variable threshold concept lattices by removing unnecessary attributes and objects without changing the lattice structure. Wang et al. [40] have provided definitions and methods for attribute reduction at local and elementary granularities in threeway concept lattices. By incorporating new levels with existing global granularity reduction, they have established a framework for trigranularity attribute reduction in threeway concept lattices. In [41], the lattice structure in generalized onesided formal context has been analyzed by defining a pair of new adjoint mappings, and attribute reduction has been computed to discuss attribute features. On the other hand, in order to simplify the calculation mechanism of FCA in formal decision context, Lin et al. [42, 43] have adopted a new matrix method to represent the intent and extent of crispfuzzy concepts in fuzzy formal context, and then presented a granular matrixbased reduction approach in consistent formal decision context. Taking advantage of graph structure, Janostik et al. [44] have proposed a general framework of addressing attribute reduction in multiple consistent formal decision contexts. However, in practical problems, not all formal decision contexts are consistent, thus, Xu et al. [45] have defined distribution attribute reduction and maximum distribution reduction set based on the congruence relation of object power sets.
In fact, the problem of uncertainty exists widely in various areas of social life. Employing linguistic information to describe the essence of things is more in line with the ambiguity and uncertainty of human thinking. Linguistic term set is suitable for expressing and dealing with linguistic decisionmaking problems in uncertain environments. Taking full advantage of linguistic term sets, multiple uncertainty linguistic representation models have been proposed, such as 2tuple fuzzy linguistic representation model [46], virtual linguistic term set [47], hesitant fuzzy linguistic term set [48], probabilistic linguistic term set [49] and so on. In a group context, solving linguistic group decisionmaking problems is also crucial. Ji et al. [50] have explored the impact of overlapping social trust relationships on consensus in social network group decision making (SNGDM) and proposed an overlapping communitydriven feedback mechanism to improve consensus. Sun et al. [51] have presented a novel framework for addressing noncooperative behavior within subgroups in largescale group decisionmaking processes. By minimizing adjustment costs and optimizing penalty parameters, this method effectively prevents excessive penalization, thereby enhancing the efficiency of collaborative decisionmaking within the group. Additionally, evaluative linguistic expressions are commonly used to describe whether an object has an attribute. Concept lattices have proven effective in handling qualitatively expressed events. Building upon this idea, Zou et al. [52] have proposed linguistic concept lattice and studied a knowledge reduction method by integrating linguistic term set into the formal context. However, their study primarily focused on determining whether objects jointly possessed attribute evaluated in qualitative language, overlooking other crucial relationships between objects and attributes. In contrast, previous research on threeway concept lattices has primarily concentrated on the presence or absence of attributes in objects, without considering the broader objectattribute relationships. For instance, in a decisionmaking problem for a company’s project assignment, it is not enough to know if employees have a particular skill or which one can achieve the best results, we may also want to know which tasks can be performed by a team of these individuals and what level they can achieve in completing the task, etc. Employees with varying skill levels can contribute differently to the overall benefits of the organization. From this point of view, it is very necessary to integrate possibility operators and necessity operators into the linguistic concept formal context. To maximize the company’s benefits, it becomes imperative to identify the optimal combination of skills and employees. During this process, we still have several challenges need to be addressed:

Dealing with uncertainty: In the real world, numerous problems involve a wealth of uncertain information, often manifested in the form of linguistic values. Traditional formal contexts can describe binary relationships between objects and attributes, but they struggle to capture linguistic information present in uncertain contexts. This is especially true in fuzzy linguistic environments, where complex associations between objects and attributes may exist. Hence, it is imperative to analyze the uncertainty information that is challenging to describe in a fuzzy linguistic environment.

Integrating possibility and necessity operators: previous studies have only considered whether objects jointly possess attributes evaluated in qualitative language or attributes, ignoring other relations between objects and attributes in formal context with linguistic data. Therefore, it becomes imperative to integrate possibility operators and necessity operators into the formal context with linguistic data for solving optimization decisionmaking problems.

Efficient knowledge reduction from big data: In the era of big data, a formidable challenge arises from the high time complexity associated with constructing a concept lattice, an NPhard problem. This complexity amplifies the difficulty of employing concept lattice methodologies for knowledge extraction, demanding careful optimization to mitigate resourceintensive processes.
The objective of this paper is to explore innovative approaches that can effectively address the challenges associated with efficient knowledge extraction from complex datasets. The main contributions are as follows:

Attributeinduced threeway objectoriented linguistic concept lattice: the proposal of an innovative attributeinduced threeway objectoriented linguistic concept lattice model that integrates possibility and necessity operators into the linguistic concept formal context to describe threeway concept lattices and linguistic concept lattices from different perspectives.

Object reduction methods: To reduce the complexity of the data, this paper proposes five object reduction methods from the perspectives of preserved lattice structure, granular concept, meetirreducible element, joinirreducible element, and equivalence relations.

Decisionmaking optimization implementation: This paper utilizes the proposed object reduction methods to analyze the constructed conceptual knowledge representation model. The simplified concept structures obtained are then applied to specific decision problems to achieve optimal decision results swiftly, thereby facilitating decisionmaking optimization.
The rest of this paper is organized as follows. Section 2 reviews some basic notions on concept lattice, threeway concept lattice, linguistic term set and linguistic concept lattice. In Section 3, we first construct an attributeinduced threeway objectoriented linguistic concept lattice, then we propose a granular reduction method and discuss the relations between five types of object reductions, including lattice reduction, meet (join)irreducible element preserving reduction, granular reduction and classification reduction based on the proposed concept lattice. The effectiveness of these object reduction methods is proved by several examples. Finally, we conclude the paper with future directions in Section 4.
2 Preliminaries
In this section, we briefly review some basic definitions and notations related to our study.
2.1 Concept lattice
Definition 1
[1] A formal context is a triple \(K=(G, M, I)\), where G is a nonempty finite set of objects, M is a nonempty finite set of attributes, and \(I\subseteq {G\times M} \) is called an incidence relation. \(I(x,a)=1\) denotes that the object x has the attribute a, and \(I(x,a)=0\) denotes that object x does not have the attribute a.
Definition 2
[8] Let \(K=(G, M, I)\) be a formal context. Given \(X\subseteq {G}\) and \(B\subseteq {M}\), a pair of operators, \(\prime :P(G)\rightarrow {P(M)}\) and \(*:P(M)\rightarrow {P(G)}\), are defined by:
If \(X^{\prime }=B\) and \(B^{*}=X\), then we call (X, B) a formal concept. Here, X is called an extent and B is called an intent of the concept (X, B).
The set of all the formal concepts forms a complete lattice called concept lattice which denoted by L(G, M, I). For any \((X_{1}, B_{1}), (X_{2}, B_{2})\in {L(G, M, I)}\), the partial order relation \("\le "\), infimum and supremum are given as follows:
Definition 3
[8] Let \(K=(G, M, I)\) be a formal context. Given \(X\subseteq {G}\) and \(B\subseteq {M}\), a pair of negative operators, \(\overline{*}:P(G)\rightarrow {P(M)}\) and \(\overline{*}:P(M)\rightarrow {P(G)}\), are defined by,
Here, \(I^{c}=(G\times {M})I\). If \(X^{\overline{\prime }}=B\) and \(B^{\overline{*}}=X\), then we call (X, B) an Nconcept.
The notation NL(G, M, I) represents a complete lattice consisting of all the Nconcepts. Within this lattice, the partial order relations, as well as the infimum and supremum operations, exhibit similarities to those in L(G, M, I).
2.2 Attributeinduced threeway concept lattice
Inspired by the idea of threeway decision, Qi et al. [8] have put forward threeway concept lattices in terms of both objects and attributes, where the attributeinduced threeway concept lattice is defined as follows.
Definition 4
[8]. Let \(K=(G, M, I)\) be a formal context. Given \(X\subseteq {G}\) and \(B\subseteq {M}\), a pair of attributeinduced threeway operators \("\lessdot "\) and \("\gtrdot "\), are defined by: \(B^{\lessdot }=(B^{*}, B^{\bar{*}})\), \((X,Y)^{\gtrdot }=\{a\in {M}a\in {X^{\prime }}\,\, and \,\, {a\in {Y^{\bar{\prime }}}}\}=X^{\prime }\cap {Y^{\bar{\prime }}}\).
Definition 5
[8]. Let \(K=(G, M, I)\) be a formal context. A pair ((X, Y), B) of an attribute subset \(B\subseteq {M}\) and two object subsets \(X,Y\subseteq {G}\) is called an attributeinduced threeway concept, or an AEconcept for concept, of (G, M, I), if and only if \(B^{\lessdot }=(X,Y)\) and \((X,Y)^{\gtrdot }= B\). (X, Y) is called the extent and B is called the intent of the AEconcept ((X, Y), B).
The AEconcepts \(((X_{1},Y_{1}),B_{1})\) and \(((X_{2},Y_{2}),B_{2})\) are ordered by:
Further, all AEconcepts form a complete lattice which called an attributeinduced threeway concept lattice(AElattice), denoted by AEL(G, M, I).
2.3 Linguistic term set
In complex environments, due to the fact that some problems are difficult to be concretively expressed by fuzzy sets, Zadeh [53] has proposed linguistic variables with evaluative linguistic expressions such as “good", “very good", “bad" and “very bad". In order to reduce the complexity of defining a grammar, Herrera et al. [54] have proposed a linguistic term set which directly supplies the term set by considering all terms as primary ones. Let \(S=\{s_{\alpha }\alpha =0,1,2,\ldots ,g\} \) be linguistic term set composed of an odd number of linguistic terms, where \(s_{\alpha }\) represents a possible value for a linguistic variable, \(g+1\) is the cardinality of the linguistic term set. Any linguistic term set must have the characteristics as follows:

1.
order relation: \(s_{\alpha }\ge s_{\beta }\), if \(\alpha \ge \beta \),

2.
negation operator: \(Neg(s_{\alpha })=s_{\beta }\), where \(\beta =g\alpha \),

3.
maximization operator: max\(\{s_{\alpha }, s_{\beta }\}=s_{\alpha }\), if \(\alpha \ge {\beta }\),

4.
minimization operator: min\(\{s_{\alpha }, s_{\beta }\}=s_{\beta }\), if \(\alpha \ge {\beta }\).
For example, a linguistic term set with five terms can be defined as \(S=\{\) \(s_{0}=\)very poor, \(s_{1}=\)poor, \(s_{2}=\)medium, \(s_{3}=\)good, \(s_{4}=\)very good}.
2.4 Linguistic concept lattice
In order to describe the relation between objects and attributes evaluated in qualitative language, Zou et al. [52] have proposed a linguistic concept formal context by introducing linguistic term set into formal concept analysis as follows.
Let’s consider a scenario where we have a set of linguistic terms denoted by \(S = \{s_{\alpha }\alpha ={0, 1, 2,. . ., g}\}\) and a set of attributes represented by \(L = \{l^{j}j={1, 2, 3, . . .,}\) \( {m}\}\), we can generate a set \(L_{S} = \{l^{j}_{s_{\alpha }}j={1,2,3,...,m}, \alpha ={0, 1, 2,. . ., g}\}\) defined on S and L, which is referred to as the linguistic concept set. This set captures the concept that an attribute l can be evaluated using any of the linguistic terms in S. For instance, if we consider the linguistic term \(s_{0}\), a corresponding linguistic concept \(l_{s_{0}}\) would indicate that attribute l is evaluated as \(s_{0}\). By utilizing this linguistic concept set, we can create a linguistic concept formal context as follows.
Definition 6
[52] A linguistic concept formal context is defined as a triple \((G, L_{S},\) \(\mathfrak {I})\), where \(G = \{x_{i}i ={1, 2, 3, . . ., n}\}\) is a nonempty finite object set, \(L_{S} = \{l^{j}_{s_{\alpha }}j={1,2,3,...,m}, \alpha ={0, 1, 2,. . ., g}\}\) is a nonempty finite linguistic concept set. \(\mathfrak {I}\) is the binary relationship from G to \(L_{S}\), i.e., \(\mathfrak {I} \subseteq {G \times { L_{S}}}\). \((x_{_{i}}, l^{j}_{s_{\alpha }})\in {\mathfrak {I}}\) indicates that the object \(x_{i}\) can be evaluated as \(s_{\alpha }\) for attribute \(l^{j}\), and \((x_{i}, l^{j}_{s_{\alpha }})\notin {\mathfrak {I}}\) indicates that the object \(x_{i}\) can not be evaluated as \(s_{\alpha }\) for attribute \(l^{j}\).
Definition 7
[52] Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context. For \(X\subseteq {G}\) and \(B_{S}\subseteq {L_S}\), we define the operators ”\({}^{\prime }\)” and ”\({}^{*}\)” as follows:
If \(X^{\prime } = B_{S}\) and \(B_{S}^{*} = X\), then a pair \((X, B_{S})\) is called a linguistic concept knowledge.
Denote all the linguistic concept knowledge of \((G, L_{S}, \mathfrak {I})\) by \(LL(G, L_{S}, \mathfrak {I})\), then \((LL(G, L_{S}, \mathfrak {I}),\) \(\le )\) forms a complete lattice, which is referred to as a linguistic concept lattice.
The partial order relation \("\le "\), infimum and supremum on linguistic concept knowledge \((X_{1}, B^{1}_{S})\) and \((X_{2}, B^{2}_{S})\) are given by:
Example 1
Suppose that Table 1 shows a linguistic concept formal context \((G, L_{S}, \mathfrak {I})\) of a company, where object set \(G=\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\}\) represents five employees, attribute set \(L=\{a, b, c\}\) represents three skills, aleadership skill, bprofessional knowledge skill and ccommunication skill. The linguistic term set for evaluating a, b, c is \(S=\{s_{0}=\)low\(, s_{1}=\)medium\(, s_{2}=\)high\(\}\), then the generated linguistic concept set is \(L_{S}=\{{a}_{{s_{0}}}, a_{{s_{1}}}, a_{{s_{2}}}\), \( b_{{s_{0}}}\), \(b_{{s_{1}}}, b_{{s_{2}}}, c_{{s_{0}}}, c_{{s_{1}}}, c_{{s_{2}}}\}\), where \({a}_{{s_{0}}}\) represents that attribute a is evaluated as \(s_{0}\), i.e., the leadership skill is low. \(\mathfrak {I}(x_{1}, a_{{s_{1}}})=1\) means that the leadership skill of employee \(x_{1}\) is medium.
By finding all possible pairs of \(X\subseteq {G}\) and \(B_S\subseteq {L_{S}}\) that satisfy above conditions, we can find \(LL(G, L_{S}, \mathfrak {I})=\{(x_{1}, a_{s_{2}}b_{s_{1}}c_{s_{2}}), (x_{2}, a_{s_{0}}b_{s_{1}}c_{s_{2}})\), \((x_{3}x_{4}, a_{s_{0}}b_{s_{2}}c_{s_{1}})\), \((x_{5}, a_{s_{2}}b_{s_{1}}\) \(c_{s_{1}})\), \((x_{1}x_{2}\), \(b_{s_{1}}c_{s_{2}})\), \((x_{1}x_{5}\), \(a_{s_{2}}b_{s_{1}})\), \((x_{2}x_{3}x_{4}, a_{s_{0}})\), \((x_{3}x_{4}x_{5},\) \(c_{s_{1}}),\) \( (x_{1}x_{2}x_{5}\), \(b_{s_{1}})\), \((G, \emptyset ), (\emptyset , L_{S})\}\). Here, for simplicity, we use \(x_{1}x_{5}\) to represent \(\{x_{1},x_{5}\}\). For instance, \((x_{1}x_{5}, a_{s_{2}}b_{s_{1}})\) is actually expressed as \((\{x_{1},x_{5}\}, \{a_{s_{2}},b_{s_{1}}\})\), and the linguistic concept knowledge \((x_{1}x_{5}, a_{s_{2}}b_{s_{1}})\) indicates that both employee \(x_{1}\) and employee \(x_{5}\) have high level of leadership skill and medium level of professional knowledge skill. We construct a Hasse diagram of linguistic concept lattice \(LL(G, L_{S}, \mathfrak {I})\) as shown in Fig. 1.
3 Object reductions of attributeinduced threeway objectoriented linguistic concept lattice
In FCA, the construction and reduction of lattices hold significant importance as research topics. In this section, our focus lies on tackling decisionmaking problems that involve attribute values expressed in linguistic terms. To address this, we study the threeway linguistic concept lattice structure and reduction approaches which are generated by modalstyle approximate operators in a linguistic concept formal context.
3.1 The construction of attributeinduced threeway objectoriented linguistic concept lattice
Definition 8
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, for \(X\subseteq {G}\) and \(B_{S}\subseteq {L_{S}}\), the operators are defined as follows:
For the sake of simplicity, we use \(x^{\prime }\) instead of \(\{x\}^{\prime }\), and \(l_{s}^{*}\) instead of \(\{l_{s}\}^{*}\). Similarly, subsequent expressions will also omit the explicit use of set notation.
Definition 9
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, \(\mathfrak {I}^{c}=G\times {L_{S}}\mathfrak {I}\). For \(X\subseteq {G}\) and \(B_{S}\subseteq {L_{S}}\), the operators are defined as follows:
Definition 10
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context. For \(X, Y\subseteq {G}\) and \(B_{S}\subseteq {L_{S}}\), the attributeinduced threeway objectoriented linguistic operators (AEOLoperators) are defined by \(B_{S}^{\unrhd }=(B_{S}^{\diamond },B_{S}^{\overline{\diamond }})\) and \((X, Y)^{\unlhd }=X^{{\Box }}\cap {Y}^{\overline{\Box }}\), respectively.
Definition 11
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context. For \(X,Y\subseteq {G}\) and \(B_{S}\subseteq {L_{S}},\) a pair \(((X, Y),B_{S})\) is called an attributeinduced threeway objectoriented linguistic concept, for short, an AEOLconcept of (G, \( L_{S}, \mathfrak {I})\) if satisfy \(B_{S}=(X,Y)^{\unlhd }\) and \(B_{S}^{\unrhd }=(X, Y)\), where (X, Y) is the extent and \(B_{S}\) is the intent of the AEOLconcept.
For two AEOLconcepts \(((X_{1}, Y_{1}),B^{1}_{S})\) and \(((X_{2}, Y_{2}),\) \(B^{2}_{S}),\) the partial order relation between them is defined as
where \((X_{1}, Y_{1})\subseteq {(X_{2}, Y_{2})}\Leftrightarrow {X_{1}\subseteq {X_{2}}}\Leftrightarrow {Y_{1}\subseteq {Y_{2}}}\), then \(((X_{1}, Y_{1}),B^{1}_{S})\) is called a subconcept of \(((X_{2}, Y_{2}),B^{2}_{S})\), and \(((X_{2}, Y_{2}),B^{2}_{S})\) is called a superconcept of \(((X_{1}, Y_{1}),B^{1}_{S})\).
For linguistic concept subset \(B_{S}\subseteq {L_{S}}\), the AEOLoperators can divide the object set G into three regions, \(POS_{B_{S}}=GB_{S}^{\overline{\diamond }}\), \(NEG_{B_{S}}=GB_{S}^{\diamond }\) and \(BND_{B_{S}}=B_{S}^{\diamond }\cap {B_{S}^{\overline{\diamond }}}\), where \(POS_{B_{S}}\) is a positive region, indicating that every object is possessed by \(B_{S}\); \(NEG_{B_{S}}\) is a negative region, indicating that every object is not possessed by \(B_{S}\); and \(BND_{B_{S}}\) is a boundary region, indicating the objects that are not sure whether they are possessed by \(B_{S}\), i.e., the objects that are neither in the positive nor in the negative region. The above analysis also shows that the AEOLconcepts can express the meaning of the attributeinduced threeway concepts.
Similarly, we can obtain the properties corresponding to attributeinduced threeway objectoriented linguistic operators in linguistic concept formal context as follows:
Proposition 1
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context. For any \( X, X_{1}\), \(X_{2},\) \(Y, Y_{1}\),\( Y_{2}\subseteq {G}, B_{S}, B_{S}^{1}, B_{S}^{2}\subseteq {L_{S}}\), the following hold:

1.
\((X_{1}, Y_{1})\subseteq {(X_{2}, Y_{2})}\Rightarrow {(X_{1}, Y_{1})^{\unlhd }\subseteq {(X_{2}, Y_{2})^{\unlhd }}}\), \(B_{S}^{1}\subseteq {B_{S}^{2}}\Rightarrow {{B_{S}^{1\unrhd }}\subseteq {{B_{S}^{2\unrhd }}}}\),

2.
\((X, Y)\subseteq {(X, Y)^{\unlhd \unrhd }}\), \(B_{S}\subseteq {{B_{S}^{\unrhd \unlhd }}}\),

3.
\((X, Y)^{\unlhd }=(X, Y)^{\unlhd \unrhd \unlhd }\), \(B_{S}^{\unrhd }=B_{S}^{\unrhd \unlhd \unrhd }\),

4.
\((X, Y)\subseteq {B_{S}^{\unrhd }}\Leftrightarrow {(X, Y)^{\unlhd }\subseteq {B_{S}}}\),

5.
\(((X_{1}, Y_{1})\cup {(X_{2}, Y_{2})})^{\unlhd }=(X_{1}, Y_{1})^{\unlhd }\cup {(X_{2}, Y_{2})^{\unlhd }}\), \((B_{S}^{1}\cup {B_{S}^{2}})^{\unrhd }=B_{S}^{1^{\unrhd }}\cup {B_{S}^{2^{\unrhd }}}\),

6.
\(((X_{1}, Y_{1})\cap {(X_{2}, Y_{2})})^{\unlhd }=(X_{1}, Y_{1})^{\unlhd }\cap {(X_{2}, Y_{2})^{\unlhd }}\), \((B_{S}^{1}\cap {B_{S}^{2}})^{\unrhd }=B_{S}^{1^{\unrhd }}\cap {B_{S}^{2^{\unrhd }}}\).
The set of AEOLL\((G, L_{S}, \mathfrak {I})\) containing all AEOLconcepts is called an attributeinduced threeway objectoriented linguistic concept lattice of \((G, L_{S}, \mathfrak {I})\), which is a complete lattice. Given two AEOLconcepts \(((X_{1}, Y_{1}),B^{1}_{S})\) and \(((X_{2}, Y_{2}),B^{2}_{S})\), the supremum and infimum are defined as follows:
Theorem 2
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, for any \(((X, Y),B_{S})\) \(\in {AEOLL(G, L_{S}, \mathfrak {I})}\), we have \(X= \bigcup \limits _{l_{s}\in {L_{S}}}l_{s}^{\diamond }\), \(Y= \bigcup \limits _{l_{s}\in {L_{S}}}l_{s}^{\overline{\diamond }}\) and \(B_{S}= \bigcup \limits _{l_{s}\in {L_{S}}}(l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }}})\).
Proof
For any \(((X, Y),B_{S})\) \(\in {AEOLL(G, L_{S}, \mathfrak {I})}\), we can get \(X=B_{S}^{\diamond }\), \(Y= B_{S}^{\overline{\diamond }}\) and \(B_{S}=X^{\Box }\cap {Y^{\overline{\Box }}}\). Then, \(X=B_{S}^{\diamond }=(\bigcup \limits _{l_{s}\in {B_{S}}}l_{s})^{\diamond }=\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\diamond }\), \(Y=B_{S}^{\overline{\diamond }}=(\bigcup \limits _{l_{s}\in {B_{S}}}l_{s})^{\overline{\diamond }}=\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\overline{\diamond }}\). \(\forall l_{s}\in {L_{S}}\), we have \(l_{s}\in {l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }}}}\), which means that \(B_{S}\subseteq {\bigcup \limits _{l_{s}\in {B_{S}}}(l_{s}^{\diamond \Box }\cap }\) \({l_{s}^{\overline{\diamond }\overline{\Box }}})\). In addition, \(B_{S}=B_{S}^{\diamond \Box }\cap {B_{S}^{\overline{\diamond }\overline{\Box }}}=(\bigcup \limits _{l_{s}\in {B_{S}}}l_{s})^{\diamond \Box }\cap {(\bigcup \limits _{l_{s}\in {B_{S}}}l_{s})^{\overline{\diamond }\overline{\Box }}} =(\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\diamond })^{\Box }\cap {(\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\overline{\diamond }})^{\overline{\Box }}}\), since \(l_{s}^{\diamond }\subseteq {\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\diamond }}\), \(l_{s}^{\overline{\diamond }}\subseteq {\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\overline{\diamond }}}\), it follows that \(l_{s}^{\diamond \Box }\subseteq {(\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\diamond })^{\Box }}\), \(l_{s}^{\overline{\diamond }\overline{\Box }}\subseteq {(\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\overline{\diamond }})^{\overline{\Box }}}\), that is, \(l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }}}\subseteq {({\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\diamond }})^{\Box }\cap {(\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\overline{\diamond }}})^{\overline{\Box }}}\). Therefore, \(\bigcup \limits _{l_{s}\in {B_{S}}}(l_{s}^{\diamond \Box }\) \({\cap {l_{s}^{\overline{\diamond }\overline{\Box }}})\subseteq {({\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\diamond }})^{\Box }\cap {(\bigcup \limits _{l_{s}\in {B_{S}}}l_{s}^{\overline{\diamond }}})^{\overline{\Box }}}}=B_{S}\). In conclusion, \(B_{S}= \bigcup \limits _{l_{s}\in {B_{S}}}(l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }}})\). \(\square \)
The study mentioned in Example 1 has expressed whether the objects jointly possessed linguistic concept. However, in a specific formal context, it is not enough to simply establish if an object has a specific attribute or linguistic concept. It is equally important to consider other perspectives and relationships between objects and attributes. Similarly, in a decisionmaking problem for project assignments within a company, it is insufficient to solely focus on an employee’s individual skills or identifying the employee who can achieve the best results. It is also essential to explore which tasks can be accomplished by a team of these individuals and what level they can collectively achieve in completing those tasks, etc. Taking these factors into account, it becomes essential to discuss an attributeinduced threeway objectoriented linguistic concept lattice as follows.
Example 2
Let us continue considering the linguistic concept formal context \((G, L_{S}, \mathfrak {I})\) as shown in Table 1. The corresponding AEOLconcepts are shown in Table 2 and the lattice structure of \(AEOLL(G, L_{S}, \mathfrak {I})\) is depicted by Fig. 2.
Among them, the node \(11\#((x_{2}x_{3}x_{4}x_{5}, x_{1}x_{2}x_{5}), a_{s_{0}}b_{s_{2}}c_{s_{1}})\) has two practical implications in terms of positive information. Firstly, it signifies that employees \(\{x_{2},x_{3},x_{4},x_{5}\}\) as a whole are able to master the skills a, b and c at the levels of \(s_{0}\), \(s_{2}\) and \(s_{1}\), respectively. In other words, these four employees are the only ones capable of achieving the above skill levels as a cohesive unit. Conversely, it implies that employee \(x_{1}\), apart from \(\{x_{2},x_{3},x_{4},x_{5}\}\), does not necessarily correspond to \(s_{0}\), \(s_{2}\) and \(s_{1}\) for the skills a, b and c. Secondly, at least one employee in \(\{x_{2},x_{3},x_{4},x_{5}\}\) can reach at least one of the three skill levels of \(\{a_{s_{0}}, b_{s_{2}}, c_{s_{1}}\}\). Analogously, regarding the negative information, the employee corresponding to skill levels of a, b and c which may not be \(s_{0}, s_{2}\) and \(s_{1}\) is object set \(\{x_{1},x_{2},x_{5}\}\), conversely, except for the objects set \(\{x_{1},x_{2},x_{5}\}\), the three skills levels of a, b, c of objects \(x_{3}\) and \(x_{4}\) must be \(s_{0}, s_{2}\) and \(s_{1}\). At the same time, at least one employee in \(\{x_{1},x_{2},x_{5}\}\) is evaluated for these three skill levels that are not at least one of \(\{a_{s_{0}}, b_{s_{2}}, c_{s_{1}}\}\).
Suppose that the company has a project that needs to be assigned to these five employees. In order to save human and material resources, the most suitable employee or team will be selected from these five employees. For instance, if the leader would like to assign the project to employees with high leadership skill, high professional knowledge skill and medium communication skill, we only need to search for the smallest AEOLconcept whose intent contains \(\{a_{s_{2}}, b_{s_{2}}, c_{s_{1}}\}\) in Table 2, i.e., \(14\#((x_{1}x_{3}x_{4}x_{5}, G), a_{s_{1}}a_{s_{2}}b_{s_{0}}b_{s_{2}}c_{s_{0}}c_{s_{1}})\) is required. The levels associated with leadership skill, professional knowledge skill and communication skill, completed by the team of \(\{x_{1},x_{3},x_{4},x_{5}\}\), can be reached at least high, high and medium, respectively. As can be seen from the negative information, none of the employees are evaluated as high leadership skill, high professional knowledge skill and medium communication skill. In other words, if we look for the team with three skill levels limited to high leadership skill, high professional knowledge skill and medium communication skill, only the team of employees \(\{x_{1},x_{3},x_{4},x_{5}\}\) can meet the requirements. If the leader would like to identify the employee with high professional knowledge skill and medium communication skill, the smallest AEOLconcept \(((x_{3}x_{4}x_{5},x_{1}x_{2}x_{5}), b_{s_{2}}c_{s_{1}})\) whose intent contains \(\{b_{s_{2}}, c_{s_{1}}\}\) can be found. The employees \(x_{3},x_{4}\) and \(x_{5}\) make up a team which at least has high professional knowledge skill and medium communication skill. Whereas from the negative information, except for employees \(x_{1},x_{2}\) and \(x_{5}\), both employees \(x_{3}\) and \(x_{4}\) can achieve high professional knowledge skill and medium communication skill. Therefore, selecting either employee \(x_{3}\) or \(x_{4}\) would fulfill the requirement of high professional knowledge and medium communication skills. This highlights the importance of not only individual employee skills but also effective teamwork in a successful company. While classical concept lattice describes whether employees jointly possess a skill in a formal context, AEOLconcepts provide a more comprehensive view by reflecting individual skill levels and teamwork levels of employees from both positive and negative aspects.
Through the above discussion, it becomes evident that the \(AEOLL(G, L_{S}, \mathfrak {I})\) not only reveals the relationships of jointly possessed and jointly nonpossessed between objects and linguistic concepts but also captures the complement relations of groups formed by these objects.
3.2 The granular reduction of AEOL lattice
Granular reduction has been widely used in formal concept analysis because of its simplicity and effectiveness. In order to preserve the granular concept of AEOL lattice while simplifying the huge background information, in this subsection, we present a method of granular reduction and study the properties of core object set and consistent set based on AEOLconcepts.
Definition 12
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, \(\forall l_{s}\in {L_{S}}\), the granular concept of AEOLconcept is defined by \(((l_{s}^{\diamond },l_{s}^{\overline{\diamond }}),l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }}})\).
Definition 13
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context. For any \(l_{{s}}\in {L_{S}}\), an object subset \(Z\subseteq {G}\) is referred to as an attributeinduced threeway objectoriented linguistic granular (AEOLG) consistent set if \(l_{{s}}^{\unrhd G\unlhd G}\) \(=l_{{s}}^{\unrhd Z\unlhd Z}\). If there is no proper object subset \(W\subset {Z}\) such that W is an AEOLG consistent set, then Z is called an AEOLG reduction of \((G, L_{S}, \mathfrak {I})\).
Theorem 3
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, for any \(l_{{s}}\in {L_{S}}\), \(Z\subseteq {G}\), \((l_{{s}}^{\diamond \Box }\cap {l_{{s}}^{\overline{\diamond }\overline{\Box }}})=(l_{{s}}^{\diamond Z\Box Z}\cap {l_{{s}}^{\overline{\diamond } Z\overline{\Box } Z}})\) if and only if Z is an AEOLG consistent set.
Proof
It follows immediately from Definition 13. \(\square \)
Obviously, when the subset Z satisfies the AEOLG consistency criterion, the positive and negative domains of the linguistic concept ls remain unchanged when restricted to Z. This criterion also provides a standard for assessing the consistency of subsets in linguistic concept formal context.
Let \(\mathcal {K}=(G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, denote all AEOLG reductions by \(Red(\mathcal {K}_{O})\). We can divide the object set into three parts according to \(Red(\mathcal {K}_{O})\) as follows:

1.
core object set \(C_{r}\): \(C_{r}=\cap {Red(\mathcal {K}_{O})}\),

2.
relatively necessary object set \(R_{r}\): \(R_{r}=\cup {Red(\mathcal {K}_{O})}\cap {Red(\mathcal {K}_{O})}\),

3.
unnecessary object set \(U_{r}\): \(U_{r}=G\cup {Red(\mathcal {K}_{O})}\).
Definition 14
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context. For any \(p, q\in {L_{S}}\), we define
where \( DIS_{AEOLG}(p,q)\) is called the AEOLG discernibility object set of \((G, L_{S}, \mathfrak {I})\).
Denote \(\Lambda _{AEOLG}=(DIS_{AEOLG}(p,q))\) as the AEOLGdiscernibility matrix, where \(\Lambda \) represents the nonempty object set of AEOLGdiscernibility matrix, then we propose the AEOLGdiscernibility function.
Definition 15
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, the AEOLGdiscernibility function is defined as follows.
where \((p^{\diamond }q^{\diamond })\bigcup ({p^{\overline{\diamond }}q^{\overline{\diamond }}})\ne \emptyset \).
In the following, we propose several theorems related to possibility and necessity operators in linguistic concept formal context.
Theorem 4
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, \(Z\subseteq {G}\). For any \(l_{s}\in {L_{S}}\), it follows that \((l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }})}\subseteq {(l_{s}^{\diamond Z\Box Z}\cap {l_{s}^{\overline{\diamond }Z\overline{\Box } Z})}}\).
Proof
For any \(b_{s}\in {L_{S}}\), suppose that \(b_{s}\in {(l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }}})}\), we can get \(b_{s}^{\diamond }\subseteq {l_{s}^{\diamond }}\), which implies that \(b_{s}^{\diamond Z}=b_{s}^{\diamond }\cap {Z}\subseteq {l_{s}^{\diamond }\cap {Z}}=l_{s}^{\diamond Z}\). Then \(b_{s}\in {l_{s}^{\diamond Z\Box Z}}\), likewise, \(b_{s}\in {l_{s}^{\overline{\diamond } Z\overline{\Box } Z}}\). Therefore, \(b_{s}\in {(l_{s}^{\diamond Z\Box Z}\cap {l_{s}^{\overline{\diamond } Z\overline{\Box } Z}})}\). Thus, we conclude \((l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }})}\subseteq {(l_{s}^{\diamond Z\Box Z}\cap {l_{s}^{\overline{\diamond }Z\overline{\Box } Z})}}\). \(\square \)
Theorem 5
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, \(Z\subseteq {G}\), Z is an AEOLG consistent set if and only if for any \(p,q\in {L_{S}}\), if \((p^{\diamond }q^{\diamond })\cup {(p^{\overline{\diamond }}q^{\overline{\diamond }}})\ne {\emptyset }\), then \(Z\cap {((p^{\diamond }q^{\diamond })\cup {(p^{\overline{\diamond }}q^{\overline{\diamond }}}))}\ne {\emptyset }\).
Proof
\((\Rightarrow )\) If Z is an AEOLG consistent set, for any \(p,q\in {L_{S}}\), \((p^{\diamond }q^{\diamond })\cup {(p^{\overline{\diamond }}q^{\overline{\diamond }}})\ne {\emptyset }\), then \(p^{\diamond }q^{\diamond }\ne {\emptyset }\) or \(p^{\overline{\diamond }}q^{\overline{\diamond }}\ne {\emptyset }\), which implies that \(q\notin {p^{\diamond \Box }}\) or \(q\notin {p^{\overline{\diamond }\overline{\Box }}}\), we have \(q\notin {p^{\diamond \Box }\cap {p^{\overline{\diamond }\overline{\Box }}}}\). Since Z is an AEOLG consistent set, \(q\notin {p^{\diamond \Box }\cap {p^{\overline{\diamond }\overline{\Box }}}}=p^{\diamond Z\Box Z}\cap {p^{\overline{\diamond } Z\overline{\Box }Z}}\), that is, \(q\notin {p^{\diamond Z\Box Z}}\) or \(q\notin {p^{\overline{\diamond }Z\overline{\Box }Z}}\), then \(p^{\diamond Z }q^{\diamond Z}\ne {\emptyset }\) or \(p^{\overline{\diamond }Z}q^{\overline{\diamond }Z}\ne {\emptyset }\). Therefore, \(p^{\diamond }\cap {Z}q^{\diamond }\cap {Z}\ne {\emptyset }\) or \(p^{\overline{\diamond }}\cap {Z}q^{\overline{\diamond }}\cap {Z}\ne {\emptyset }\). Thus, we obtain \(Z\cap {((p^{\diamond }q^{\diamond })\cup {(p^{\overline{\diamond }}q^{\overline{\diamond }}}))}\ne {\emptyset }\).
\((\Leftarrow )\) For any \(p,q\in {L_{S}}\), if \((p^{\diamond }q^{\diamond })\cup {(p^{\overline{\diamond }}q^{\overline{\diamond }}})\ne {\emptyset }\), then \(Z\cap {((p^{\diamond }q^{\diamond })\cup {(p^{\overline{\diamond }}q^{\overline{\diamond }}}))}\ne {\emptyset }\). To prove that Z is an AEOLG consistent set, only \(p^{\diamond \Box }\cap {p^{\overline{\diamond }\overline{\Box }}}=p^{\diamond Z\Box Z}\cap {p^{\overline{\diamond } Z\overline{\Box }Z}}\) is required. It is obviously that \(p^{\diamond \Box }\cap {p^{\overline{\diamond }\overline{\Box }}}\subseteq {p^{\diamond Z\Box Z}\cap {p^{\overline{\diamond } Z\overline{\Box }Z}}}\) according to Theorem 4, and we need to prove that \(p^{\diamond \Box }\cap {p^{\overline{\diamond }\overline{\Box }}}\supseteq {p^{\diamond Z\Box Z}\cap {p^{\overline{\diamond } Z\overline{\Box }Z}}}\). Suppose that \(q\notin {p^{\diamond \Box }\cap {p^{\overline{\diamond }\overline{\Box }}}}\), we have \(q\notin {p^{\diamond \Box }}\) or \(q\notin {p^{\overline{\diamond }\overline{\Box }}}\), that is, \(p^{\diamond }q^{\diamond }\ne {\emptyset }\) or \(p^{\overline{\diamond }}q^{\overline{\diamond }}\ne {\emptyset }\), \((p^{\diamond }q^{\diamond })\cup {(p^{\overline{\diamond }}q^{\overline{\diamond }}})\ne {\emptyset }\). Since \(Z\cap {((p^{\diamond }q^{\diamond })\cup {(p^{\overline{\diamond }}q^{\overline{\diamond }}}))}\ne {\emptyset }\), then \((p^{\diamond Z}q^{\diamond Z})\cup {(p^{\overline{\diamond }Z}q^{\overline{\diamond }Z}})\ne {\emptyset }\), which implies that \(q\notin {p^{\diamond Z\Box Z}}\) or \(q\notin {p^{\overline{\diamond }Z\overline{\Box }Z}}\), i.e., \(q\notin {p^{\diamond Z\Box Z}\cap {p^{\overline{\diamond } Z\overline{\Box }Z}}}\), therefore, \(p^{\diamond \Box }\cap {p^{\overline{\diamond }\overline{\Box }}}\supseteq {p^{\diamond Z\Box Z}\cap {p^{\overline{\diamond } Z\overline{\Box }Z}}}\). Above all, \(p^{\diamond \Box }\cap {p^{\overline{\diamond }\overline{\Box }}}={p^{\diamond Z\Box Z}\cap }\) \({{p^{\overline{\diamond } Z\overline{\Box }Z}}}\), Z is an AEOLG consistent set. \(\square \)
Theorem 6
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, for any \(x\in {G}\), \(p, q\in {L_{S}}\), x is a core object if and only if there exists \(DIS_{AEOLG}(p,q)=\{x\}\).
Proof
It can be proved directly by Definition 15. \(\square \)
Determining core objects is a crucial step in computing object granular reduction because core objects are essential for identifying and understanding the inherent relational properties within a given linguistic concept formal context. Serving as pivotal indicators, core objects facilitate the comprehension of relationships among objects, thus enabling a more efficient process of object granular reduction. By pinpointing core objects, we can focus on the most significant elements, thereby streamlining and optimizing the representation and analysis of concepts.
Within such a framework, we can provide the granular reduction algorithm in a linguistic concept formal context as follows. If there are t granular concepts in linguistic concept formal context \((G, L_{S}, \mathfrak {I})\), it’s easily evident that the time complexity of this algorithm can be computed as \(\frac{t^{2}2t+5}{4}t^{2}GL_{S}^{2}\).
Example 3
Considering the linguistic concept formal context \((G, L_{S}, \mathfrak {I})\) of a company in Table 1, we can get the granular concepts of AEOLconcepts according to Definition 12: \(((x_{2}x_{3}x_{4},x_{1}x_{5}),\) \( a_{s_{0}})\), \(((x_{1}x_{5}, x_{2}x_{3}x_{4}), a_{s_{2}})\), \(((x_{3}x_{4}, x_{1}x_{2}x_{5}), b_{s_{2}})\), \(((x_{1}x_{2}x_{5},\) \(x_{3}x_{4}), b_{s_{1}})\), \(((x_{3}x_{4}x_{5},x_{1}x_{2})\!,\!\) \(c_{s_{1}})\), \(((x_{1}x_{2},x_{3}x_{4}x_{5}), c_{s_{2}})\), \(((\emptyset , G), a_{s_{1}}b_{s_{0}}c_{s_{0}})\). Then the AEOLGdiscernibility matrix is shown in Table 3, the AEOLGdiscernibility function is calculated as follows. \(f(\Lambda _{AEOLG})=\bigwedge \bigvee \limits _{p,q\in {L_{S}}}((p^{\diamond }q^{\diamond })\bigcup ({p^{\overline{\diamond }}q^{\overline{\diamond }}}))\)
\(=x_{2}\wedge {x_{5}}\wedge (x_{3}\vee x_{4})=(x_{2}\wedge x_{3}\wedge x_{5})\vee (x_{2}\wedge x_{4}\wedge x_{5})\).
Therefore, we obtain two object granular reductions in \((G, L_{S}, \mathfrak {I})\), which are \(\{x_{2}, x_{3},\) \( x_{5}\}\) and \(\{x_{2}, x_{4}, x_{5}\}\). Let’s take the AEOLG reduction \(\{x_{2}, x_{3}\), \( x_{5}\}\) as an example. The corresponding AEOLconcepts are updated and presented in Table 4.
Continuing with the project assignment issues mentioned in the previous subsection. It is obvious that \(\{x_{2}, x_{5}\}\) is a core object set, \(\{x_{3}, x_{4}\}\) is a relatively necessary object set, and \(\{x_{1}\}\) is an unnecessary object set, we can determine that \(((x_{3}x_{5}, x_{2}x_{3}x_{5}), a_{s_{1}}a_{s_{2}}b_{s_{0}}b_{s_{2}}c_{s_{0}}c_{s_{1}})\) is the most satisfying AEOLconcept in Table 4 if the goal is to find employees with high leadership skill, high professional knowledge skill, and medium communication skill. This means that a team composed of employees \(x_{3}\) and \(x_{5}\) can effectively maximize the benefits by achieving the desired skill levels while utilizing the least number of employees. Similarly, for the AEOLconcepts under AEOLG reduction \(\{x_{2}, x_{4}, x_{5}\}\), we find that a team composed of employees \(x_{4}\) and \(x_{5}\) can also achieve high leadership skill, high professional knowledge skill, and medium communication skill. Therefore, whether the company selects a team from \(\{x_{3}, x_{5}\}\) or \(\{x_{4}, x_{5}\}\), they will be able to successfully complete the project. If the goal is to find employees with high professional knowledge skill and medium communication skill, we can also obtain the same result as the original AEOLconcepts shown in Example 2, regardless of whether the employee is \(x_{3}\) or \(x_{4}\). The granular concepts of AEOLconcepts show the object set corresponding to each linguistic concept or each skill level mastered by each employee. AEOLG reduction preserves the information of each skill level to generate new AEOLconcepts, and the combination of required skill levels can be analyzed to search for the optimal decision result with the least manpower and resources.
Finally, it is worth mentioning that an object granular reduction is the minimal consistent set preserving all the linguistic concept granule information of \(AEOLL(G, L_{S}, \mathfrak {I})\).
3.3 The relations between five types of object reductions based on AEOL lattice
Granular reduction is a data analysis method based on lattice theory, which holds significant value in the fields of data mining and knowledge discovery. Concept lattice reduction, on the other hand, is a further simplification of data representation built upon granular reduction. It transforms complex datasets into concise forms, enhancing the clarity and visibility of data structure and relationships. Through visual analysis, we gain a more intuitive understanding of the inherent patterns and characteristics of the data, enabling the discovery of hidden patterns and trends. Additionally, to streamline the structure of the concept lattice, it is also crucial to preserve the meet (join)irreducible elements and equivalence relations within a formal context involving linguistic data. Hence, in this section, we introduce lattice reduction, meet (join)irreducible element preserving reduction, and classification reduction based on linguistic concepts of AEOL lattice. Subsequently, we explore the relationships between these four types of reductions and granular reduction.
Obviously, for any ((X, Y), \(B_{S})\in {AEOLL(G, L_{S}, \mathfrak {I})}\), the set of all the intent of AEOLconcepts can be denoted by \(Int_{AEOLL}=\{B_{S}\subseteq {L_{S}}B_{S}^{\diamond \Box }\cap {B_{S}^{\overline{\diamond }\overline{\Box }}}=B_{S}\}\), thus, lattice consistent set can be defined as follows.
Definition 16
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context. An object subset \(Z\subseteq {G}\) is referred to as an attributeinduced threeway objectoriented linguistic concept lattice(AEOLL) consistent set if \(Int_{AEOLL}(G, L_{S}, \mathfrak {I})\) \(=Int_{AEOLL}(Z, L_{S}, \mathfrak {I}_{Z})\), where \(\mathfrak {I}_{Z}=\mathfrak {I}\cap {(Z\times {L_{S}})}\). If there is no proper object subset \(W\subset {Z}\) such that W is an AEOLL consistent set, then Z is called an AEOLL reduction.
Theorem 7
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, for any object subset \(Z\subseteq {G}\), if Z is an AEOLL consistent set, then Z is an AEOLG consistent set of \((G, L_{S}, \mathfrak {I})\).
Proof
It suffices to show that if \(Int_{AEOLL}(G, L_{S}, \mathfrak {I})\) \(=Int_{AEOLL}(Z,\) \(L_{S}, \mathfrak {I}_{Z})\), then for any \(l_{s}\in {L_{S}}\), it satisfies \((l_{s}^{\diamond Z\Box Z}\cap {l_{s}^{\overline{\diamond } Z\overline{\Box } Z}})={(l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }})}}\). Obviously, \((l_{s}^{\diamond Z\Box Z}\cap {l_{s}^{\overline{\diamond } Z\overline{\Box } Z}})\supseteq {(l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }})}}\) holds by Theorem 4. For any \(l_{s}^{\diamond Z\Box Z}\cap {l_{s}^{\overline{\diamond } Z\overline{\Box } Z}\in {Int_{AEOLL}(Z,L_{S}, \mathfrak {I}_{Z})}}\), there exists \( B^{1}_{S}\in {Int_{AEOLL}(Z,}\) \( L_{S}, \mathfrak {I}_{Z})\) such that \(B^{1}_{S}=l_{s}^{\diamond Z\Box Z}\cap {l_{s}^{\overline{\diamond } Z\overline{\Box } Z}}\). Then since \(l_{s}^{\diamond Z\Box Z}\cap {l_{s}^{\overline{\diamond } Z\overline{\Box } Z}}=B^{1}_{S}=\bigcup \limits _{l_{s}\in {B^{1}_{S}}}(l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }}})\), \(\forall (l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }}})\in {Int_{AEOLL}}\) \({(G, L_{S}, \mathfrak {I})}\) and \(Int_{AEOLL}(G,\) \( L_{S}, \mathfrak {I})\) \(=Int_{AEOLL}(Z,\) \(L_{S}, \mathfrak {I}_{Z})\), it must have \(l_{s}^{\diamond Z\Box Z}\cap {l_{s}^{\overline{\diamond } Z\overline{\Box } Z}}=l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }}}\), therefore, \((l_{s}^{\diamond Z\Box Z}\cap {l_{s}^{\overline{\diamond } Z\overline{\Box } Z}})\subseteq {(l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }})}}\). The above shows that \((l_{s}^{\diamond Z\Box Z}\cap {l_{s}^{\overline{\diamond } Z\overline{\Box } Z}})={(l_{s}^{\diamond \Box }\cap {l_{s}^{\overline{\diamond }\overline{\Box }})}}\), i.e., Z is an AEOLG consistent set of \((G, L_{S}, \mathfrak {I})\). \(\square \)
Corollary 1
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, \(Z\subseteq {G}\). If Z is an AEOLL reduction, then there exists an object set \(W\subseteq {Z}\) such that W is an AEOLG reduction.
Proof
Similar to the proof of Theorem 7. \(\square \)
Through the above analysis, it can be concluded that if there exists an AEOLL reduction Z in the given linguistic concept formal context \((G, L_{S}, \mathfrak {I})\), then an object set W can be found, which is a subset of Z and satisfies the conditions of an AEOLG reduction. This implies that AEOLL reduction can be further refined into an AEOLG reduction, providing more specific information in the representation and processing of linguistic concepts.
For notational simplicity, we denote \(Int_{AEOLM}(G,\) \( L_{S}, \mathfrak {I})\) as the intent set of all the meetirreducible elements of \(AEOLL(G, L_{S}, \mathfrak {I})\) and denote \(Int_{AEOLJ}(G, \) \(L_{S}, \mathfrak {I})\) as the intent set of all the joinirreducible elements of \(AEOLL(G, L_{S}, \mathfrak {I})\).
Definition 17
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context. An object subset \(Z\subseteq {G}\) is referred to as an attributeinduced threeway objectoriented linguistic meetirreducible element (AEOLMIE) preserving consistent set if \(Int_{AEOLM}(G, L_{S}, \mathfrak {I})\)= \(Int_{AEOLM}(Z, L_{S}, \mathfrak {I}_{Z})\). If there is no proper object subset \(W\subset {Z}\) such that W is an AEOLMIE preserving consistent set, then Z is called an AEOLMIE preserving reduction.
Theorem 8
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, \(Z\subseteq {G}\). Then, Z is an AEOLMIE preserving consistent set if and only if Z is an AEOLL consistent set of \((G, L_{S}, \mathfrak {I})\).
Proof
\((\Rightarrow )\) Suppose that Z is an AEOLMIE preserving consistent set, for any \(B_{s}\in {Int_{AEOLL}(G, L_{S}, \mathfrak {I})}\), there exist \(E_{S}, F_{S}\in {Int_{AEOLM}(G, L_{S}, \mathfrak {I})}\) such that \(B_{S}=E_{S}\) \(\cap {F_{S}}\). Since \(Int_{AEOLM}(G, L_{S}, \mathfrak {I})=Int_{AEOLM}(Z, L_{S},\) \(\mathfrak {I}_{Z})\), we have \(E_{S}\cap {F_{S}}=B_{S}\), then \(B_{S}\in {Int_{AEOLL}}\) \({(Z, L_{S}, \mathfrak {I}_{Z})}\), which means that \(Int_{AEOLL}(G, L_{S}, \mathfrak {I})\subseteq {Int_{AEOLL}(Z, L_{S}, \mathfrak {I}_{Z})}\). Obviously, \(Int_{AEOLL}(G, L_{S}, \mathfrak {I})\supseteq {Int_{AEOLL}(Z, L_{S}, \mathfrak {I}_{Z})}\), we can conclude that \(Int_{AEOLL}(G,\) \( L_{S}, \mathfrak {I})=Int_{AEOLL}(Z, L_{S}, \mathfrak {I}_{Z})\). Hence, Z is an AEOLL consistent set of \((G, L_{S}, \mathfrak {I})\).
\((\Leftarrow )\) Suppose that Z is an AEOLL consistent set, we can get \(Int_{AEOLL}(G, L_{S},\) \( \mathfrak {I})=Int_{AEOLL}(Z, L_{S}, \mathfrak {I}_{Z})\). For any AEOLmeetirreducible element \(((X, Y), B_{S})\), if \(B_{S}\in {Int_{AEOLM}(G, L_{S}, \mathfrak {I})}\), it follows that \(B_{S}\ne {E_{S}\cap {F_{S}}}\), where \(E_{S}, F_{S}\in {Int_{AEOLL}(G,}\) \({L_{S}, \mathfrak {I})}\) and \(E_{S}, F_{S}\ne {B_{S}}\), i.e., \(B_{S}\in {Int_{AEOLL}(Z, L_{S}, \mathfrak {I}_{Z})}\). We have \(B_{S}\in {Int_{AEOLM}(Z,}{L_{S}, \mathfrak {I}_{Z})}\) because of the AEOLmeetirreducible element. Thus, \(Int_{AEOLM}(G, L_{S}, \mathfrak {I})\subseteq {Int_{AEOLM}}\) \({(Z, L_{S}, \mathfrak {I}_{Z})}\). Similarly, it holds that \(Int_{AEOLM}(G, L_{S}, \mathfrak {I})\supseteq {Int_{AEOLM}(Z, L_{S}, \mathfrak {I}_{Z})}\), which implies that \(Int_{AEOLM}(G, L_{S}, \mathfrak {I})=Int_{AEOLM}(Z, L_{S}, \mathfrak {I}_{Z})\). Therefore, Z is an AEOLMIE preserving consistent set. \(\square \)
Corollary 2
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, \(Z\subseteq {G}\). Then, Z is an AEOLMIE preserving reduction if and only if Z is an AEOLL reduction.
Definition 18
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context. An object subset \(Z\subseteq {G}\) is referred to as an attributeinduced threeway objectoriented linguistic joinirreducible element (AEOLJIE) preserving consistent set if \(Int_{AEOLJ}(G, L_{S}, \mathfrak {I})\) \(=Int_{AEOLJ}(Z, L_{S}, \mathfrak {I}_{Z})\). If there is no proper object subset \(W\subset {Z}\) such that W is an AEOLJIE preserving consistent set, then Z is called an AEOLJIE preserving reduction.
Theorem 9
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, for any \(((X, Y), B_{S})\) \(\in {AEOLL(G, L_{S}, \mathfrak {I})}\), \(((X, Y), B_{S})\) is an attributeinduced threeway objectoriented linguistic joinirreducible element (AEOLJIE) if and only if \(((X, Y), B_{S})\) is the granular concept of AEOLconcepts.
Proof
\((\Rightarrow )\) Assume that \(((X, Y), B_{S})\) is an attributeinduced threeway objectoriented linguistic joinirreducible element, then for any intent \(B^{j}_{S}\in {L_{S}}\) of AEOLJIE, we have \(B^{j}_{S}\in {Int_{AEOLJ}(G, L_{S},}\) \({\mathfrak {I})}\) such that \(\cup {B^{j}_{S}}=C_{S}\in {Int_{AEOLL}(G, L_{S}, \mathfrak {I})}\). It means that for any AEOLconcept, we can get \((C^{\unrhd }_{S},C^{\unrhd \unlhd }_{S})=(\cup {(B^{j}_{S})}^{\unrhd },\) \(\cup {(B^{j}_{S})}^{\unrhd \unlhd })=\vee {(((B^{j}_{S})^{\diamond }, { (B^{j}_{S})^{\overline{\diamond }})}}\), \({(B^{j}_{S})^{\diamond \Box }\cap {(B^{j}_{S})} ^{\overline{\diamond }\overline{\Box }}} )\). Therefore, \(((X, Y), B_{S})\) is the granular concept of AEOLconcepts.
\((\Leftarrow )\) Analogous to the necessity, it can be held easily. \(\square \)
According to the Theorem 9, we can obtain the following Theorem and Corollary obviously.
Theorem 10
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, \(Z\subseteq {G}\). Then, Z is an AEOLJIE preserving consistent set if and only if Z is an AEOLG consistent set of \((G, L_{S}, \mathfrak {I})\).
Corollary 3
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, \(Z\subseteq {G}\). Then, Z is an AEOLJIE preserving reduction if and only if Z is an AEOLG reduction.
According to the viewpoint of rough sets, it is easy to find out the discernibility relation between the linguistic concept sets with respect to the objects, so the classification reduction method of \(AEOLL(G, L_{S}, \mathfrak {I})\) can be obtained.
Definition 19
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, for any object subset \(Z\subseteq {G}\), the binary relation on \(L_{S}\) is defined as follows:
If \(R_{Z}^{\diamond }=R_{G}^{\diamond }\), then Z is referred to as an attributeinduced threeway objectoriented linguistic classification (AEOLC) consistent set. If there is no proper object subset \(W\subset {G}\) such that \(R_{W}^{\diamond }=R_{G}^{\diamond }\), then Z is an AEOLC reduction of \((G, L_{S}, \mathfrak {I})\).
In a linguistic concept formal context \((G, L_{S}, \mathfrak {I})\), the following properties can easily hold on \(R_{Z}^{\diamond }\):

1.
if \(W\subseteq {Z}\subseteq {G}\), then \(R_{W}^{\diamond }\supseteq {R_{Z}^{\diamond }}\supseteq {R_{G}^{\diamond }}\),

2.
if \(W, Z\subseteq {G}\), then \(R_{{W}\cup {Z}}^{\diamond }=R_{W}^{\diamond }\cap {R_{Z}^{\diamond }}\).
Theorem 11
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, \(Z\subseteq {G}\). Then, Z is an AEOLG consistent set if and only if Z is an AEOLC consistent set of \((G, L_{S}, \mathfrak {I})\).
Proof
It suffices to show that \((p^{\diamond Z\Box Z}\cap {p^{\overline{\diamond } Z\overline{\Box } Z}})={(p^{\diamond \Box }\cap {p^{\overline{\diamond }\overline{\Box }})}}\) iff \(R_{Z}^{\diamond }=R_{G}^{\diamond }\) for any \(p\in {L_{S}}\).
\((\Rightarrow )\) Obviously, we have \(R_{Z}^{\diamond }\supseteq {R_{G}^{\diamond }}\). To prove that Z is an AEOLC consistent set, we only need to prove that \(R_{Z}^{\diamond }\subseteq {R_{G}^{\diamond }}\). For any \(p,q\in {L_{S}}\), if \((p,q)\notin {R_{G}^{\diamond }}\), we have \(p^{\diamond }\ne {q^{\diamond }}\) or \(p^{\overline{\diamond }}\ne {q^{\overline{\diamond }}}\), which implies \(p^{\diamond }{q^{\diamond }}\ne {\emptyset }\) or \(p^{\overline{\diamond }}{q^{\overline{\diamond }}}\ne {\emptyset }\), i.e., \(q\notin {p^{\diamond \Box }\cap {p^{\overline{\diamond }\overline{\Box }}}}\). Since Z is an AEOLG consistent set, \((p^{\diamond Z\Box Z}\cap {p^{\overline{\diamond } Z\overline{\Box } Z}})={(p^{\diamond \Box }\cap {p^{\overline{\diamond }\overline{\Box }})}}\), we obtain \(q\notin {(p^{\diamond Z\Box Z}\cap {p^{\overline{\diamond } Z\overline{\Box } Z}})}\), that is, \(p^{\diamond Z}{q^{\diamond Z}}\ne {\emptyset }\) or \(p^{\overline{\diamond } Z}{q^{\overline{\diamond } Z}}\ne {\emptyset }\), then \(p^{\diamond Z}\ne {q^{\diamond Z}}\) or \(p^{\overline{\diamond } Z}\ne {q^{\overline{\diamond } Z}}\). Therefore, \((p,q)\notin {R_{Z}^{\diamond }}\), \(R_{Z}^{\diamond }\subseteq {R_{G}^{\diamond }}\), we conclude that \(R_{Z}^{\diamond }={R_{G}^{\diamond }}\), Z is an AEOLC consistent set of \((G, L_{S}, \mathfrak {I})\).
\((\Leftarrow )\) Similarly, it can be proved according to the Definition 19. \(\square \)
Corollary 4
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, \(Z\subseteq {G}\). Then, Z is an AEOLG reduction if and only if Z is an AEOLC reduction.
Proof
It follows immediately from Theorem 11. \(\square \)
By recognizing the equivalence between the aforementioned reduction methods, at times, we can focus our efforts on studying one type of reduction without simultaneously considering multiple different scenarios. This approach allows for a more systematic and efficient exploration and comprehension of reduction properties within the linguistic concept formal context. However, specific circumstances warrant individual analysis.
Through the discernibility matrix and discernibility function, the calculation method of lattice reduction and meetirreducible element (AEOLMIE) preserving reduction can be defined as follows.
Definition 20
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context. For any ((X, Y), \(B_{S}))\), \(((Z, W), C_{S}))\in {AEOLL(G, L_{S}, \mathfrak {I})}\), we define
Where \( DIS_{AEOLL}(((X, Y), B_{S}),((Z, W), C_{S}))\) is called the AEOLLdiscernibility object set of \(((X, Y), B_{S})\) and \(((Z, W), C_{S})\).
We denote \(\Lambda _{AEOLL}=(DIS_{AEOLL}(((X, Y), B_{S})),((Z,\) \( W), C_{S}))\) as the AEOLLdiscernibility matrix, then the AEOLLdiscernibility function is proposed.
Definition 21
Let \((G, L_{S}, \mathfrak {I})\) be a linguistic concept formal context, the AEOLLdiscernibility function is defined as follows.
In this section, four object reduction methods are introduced, and their relationships are discussed. The computational processes of these reduction methods are quite similar. Below, we provide the lattice reduction algorithm for linguistic concept formal contexts, using lattice reduction as an example. Its time complexity calculation follows a methodology akin to the granular reduction algorithm and will not be reiterated.
Example 4
Considering the linguistic concept formal context \((G, L_{S}, \mathfrak {I})\) in Table 1, we can obtain the AEOLL reductions and AEOLMIE preserving reductions using the discernibility function. These reductions are \(\{x_{1}, x_{2}, x_{3}, x_{5}\}\) and \(\{x_{1}, x_{2}, x_{4}, x_{5}\}\). The AEOLJIE preserving reduction set and AEOLC reduction set are the same as AEOLG reduction set, which are \(\{x_{2}, x_{3}, x_{5}\}\) and \(\{x_{2}, x_{4}, x_{5}\}\).
The AEOLconcepts can be updated using AEOLL reduction for \(\{x_{1}, x_{2}, x_{3}, x_{5}\}\), as shown in Table 5. In order to find employees who possess high leadership skills, medium professional knowledge, and high communication skills while minimizing manpower for a company project assignment problem, we can query the concept using the following AEOLconcepts obtained from the three methods in Tables 2, 4, and 5:
AEOLconcept based on Table 2: \(((x_{1}x_{2}x_{5}, x_{2}x_{3}x_{4}x_{5}), a_{s_{2}}\) \(b_{s_{1}}c_{s_{2}})\).
AEOLconcept based on Table 4: \(((x_{2}x_{5},x_{2}x_{3}x_{5}), a_{s_{1}}a_{s_{2}}\) \(b_{s_{0}}b_{s_{1}}c_{s_{0}}c_{s_{2}})\).
AEOLconcept based on Table 5: \(((x_{1}x_{2}x_{5}, x_{2}x_{3}x_{5}), a_{s_{2}}\) \(b_{s_{1}}c_{s_{2}})\).
The negative information from concepts \(((x_{1}x_{2}x_{5}, x_{2}x_{3}x_{4}\) \(x_{5}), a_{s_{2}}b_{s_{1}}c_{s_{2}})\) and \(((x_{1}x_{2}x_{5}, x_{2}x_{3}x_{5}), a_{s_{2}}b_{s_{1}}c_{s_{2}})\) indicates that employee \(x_{1}\) can achieve high leadership skill, medium professional knowledge, and high communication skill on their own. However, concept \(((x_{2}x_{5},x_{2}x_{3}x_{5}), a_{s_{1}}a_{s_{2}}b_{s_{0}}b_{s_{1}}c_{s_{0}}\) \(c_{s_{2}})\) shows that no single employee meets these requirements except for a team consisting of employees \(x_{2}\) and \(x_{5}\). Therefore, employee \(x_{1}\) is the optimal decision for the project, analogous to the AEOLL reductions \(\{x_{1}, x_{2}, x_{4}, x_{5}\}\). Continuing with the AEOLL reduction method, let’s further discuss the decisionmaking problems in examples 2 and 3. If the leader wants to query employees with high leadership skill, high professional knowledge, and medium communication skill, we can find that employees \(\{x_{1}, x_{3}, x_{5}\}\) or \(\{x_{1}, x_{4}, x_{5}\}\) satisfy the requirement. Similarly, if the leader wants to query employees with high professional knowledge and medium communication skill, then employee \(\{x_{3}\}\) or \(\{x_{4}\}\) would be the optimal decision to complete the project.
The examples provided above regarding the optimization decisionmaking for a company in Table 1 can be summarized by referring to Fig. 3. We can conclude that while AEOLG reduction can obtain a streamlined decision result in most cases, AEOLL reduction can sometimes yield a more precise result. The AEOLconcepts obtained through AEOLL reduction express the background information in a more concise concept and contain more knowledge than those obtained through AEOLG reduction, compared to the original AEOLconcepts of \(AEOLL(G, L_{S}, \mathfrak {I})\).
Referring to reference [38], the relations of these five reduction methods can be explained by their corresponding consistent sets, as shown in Fig. 4.
From Fig. 4, we can see that if Z is an AEOLL reduction, then Z must also be reductions of AEOLMIE, AEOLJIE, AEOLG, and AEOLC. Furthermore, AEOLL reduction is equivalent to AEOLMIE reduction; hence, the necessary information for AEOLL reduction can be obtained by computing AEOLMIE reduction. Additionally, it can be observed that AEOLG reduction is equivalent to reductions of AEOLC and AEOLJIE. Therefore, maintaining joinirreducible elements can also yield AEOLG reduction results, or alternatively, analyzing the equivalence relationships corresponding to possibility operators directly from the formal context to discuss AEOLC reduction. Consequently, different reduction methods can be chosen based on practical requirements.
To further illustrate the effectiveness of the research in this section, a comparison was made with references [33, 35, 43], summarizing the characteristics of these reduction methods as shown in Table 6. Reference [33] introduced an attribute reduction method that maintains the invariance of granular concepts, referred to as LG reduction. Reference [35] proposed an attribute reduction method using the principle of decision rule invariance, denoted as \(DL_{O}\) reduction. Reference [43] proposed an attribute reduction method that preserves the invariance of granular matrices, denoted as OFTLM reduction.
According to Table 6, it is evident that both \(DL_{O}\) reduction and LG reduction are relatively simple methods but involve knowledge loss. They are effective in handling twodimensional data with 0 and 1, making them suitable for rule acquisition and granular computing problems. When dealing with fuzzy data and granular computing issues, OFTLM reduction exhibits simplicity alongside knowledge loss, effectively addressing fuzzy data in a fuzzy environment. However, these methods face challenges in reducing reallife linguistic information. The five reduction methods proposed in this paper can directly handle linguistic terms, avoiding information loss during the conversion process. Furthermore, although AEOLL reduction introduced in this paper is relatively complex, it achieves reduction without knowledge loss, preserving the integrity of the original knowledge. AEOLMIE and AEOLJIE reduction methods can effectively construct lattice structures through the meet(join)irreducible elements of AEOLL\((G, L_{S}, \mathfrak {I})\). Additionally, AEOLG and AEOLC reduction methods generate granular concepts and equivalence relations corresponding to linguistic evaluative attributes, which is faster and more effective than obtaining knowledge directly from the formal context. Therefore, employing the object reduction methods proposed in this paper in a fuzzy linguistic environment offers advantages such as no knowledge loss, simplification of complexity, precise reasoning and computation, and more accurate classification results. These methods can assist researchers and practitioners in better understanding and managing complex information in fuzzy linguistic environment, leading to improved outcomes.
4 Conclusion
The threeway decision theory is widely recognized as a practical approach for solving decisionmaking problems in uncertain environments. Knowledge reduction and granular computing are crucial components of knowledge discovery and knowledge acquisition in complex data contexts. In this paper, several object reduction methods have been proposed to tackle optimal decisionmaking problems with linguistic data. Specifically, we have introduced a granular reduction approach that combines threeway concept lattice with necessity and possibility operators. This approach ensures the preservation of the granular concept in the linguistic concept formal context. We have constructed the AEOL lattice to divide the object set into three regions, which enables us to express the jointly possessed information shown by linguistic concept lattice and the possible information possessed by the linguistic concept set. The granular concept, granular discernibility function and granular consistent set have been employed to acquire the granular reduction based on AEOL lattice. Then five types of consistent sets have been defined to discuss the relations between granular reduction, lattice reduction, meet (join)irreducible element reduction and classification reduction. Through the analysis of decisionmaking problems of different queries in project assignment within a company, the effective identification of the optimal employee or team corresponding to the AEOLconcepts can be achieved by employing different object reduction methods, which emphasizes the distinct advantages of these object reduction methods in decisionmaking and effectively showcases the importance of both individual employees and teamwork.
In conclusion, while our study has demonstrated the efficacy of some reduction methods in decisionmaking with linguistic data, it’s essential to acknowledge certain limitations. For instance, the current framework primarily focuses on crisp linguistic data, and extending it to handle fuzzy linguistic concept formal contexts remains a challenge. Furthermore, addressing knowledge acquisition in inconsistent linguistic formal decision contexts warrants further investigation. These limitations highlight avenues for future research to enhance the applicability and robustness of our proposed approach.
References
Ganter B, Wille R (2012) Formal concept analysis: mathematical foundations. Springer, Berlin. https://doi.org/10.1007/9783642598302
Belohlavek R, Baets BD, Konecny J (2014) Granularity of attributes in formal concept analysis. Inf Sci 260:149–170. https://doi.org/10.1016/j.ins.2013.10.021
Kumar P, Aswani S, Cherukuri K, Li J (2014) Concepts reduction in formal concept analysis with fuzzy setting using shannon entropy. Int J Mach Learn Cybern 8(1):179–189. https://doi.org/10.1007/s1304201403136
Yan E, Yu C, Lu L, Hong W, Tang C (2021) Incremental concept cognitive learning based on threeway partial order structure. KnowlBased Syst 5:1–11. https://doi.org/10.1016/j.knosys.2021.106898
Zhang T, Li HH, Liu MQ, Rong M (2020) Incremental conceptcognitive learning based on attribute topology. Int J Approx Reason 118:173–189. https://doi.org/10.1016/j.ijar.2019.12.010
Qian T, Yang Y, He X (2021) The feature description of formal context based on the relationships among concept lattices. Entropy (Basel, Switzerland) 23(11):2–12. https://doi.org/10.3390/e23111397
Hu M, Tsang ECC, Guo Y, Zhang Q, Chen D, Xu W (2022) A novel approach to conceptcognitive learning in intervalvalued formal contexts: a granular computing viewpoint. Int J Mach Learn Cybern(13–4):1049–1064. https://doi.org/10.1007/s13042021014341
Qi J, Wei L, Yao Y (2014) Threeway formal concept analysis. In: International conference on rough sets and knowledge technology, pp 732–741. https://doi.org/10.1007/9783319117409_67
Zhao X, Miao D, Fujita H (2021) Variableprecision threeway concepts in lcontexts. Int J Approx Reason 130:107–125. https://doi.org/10.1016/j.ijar.2020.11.005
Zhai Y, Qi J, Li D, Zhang C, Xu W (2022) The structure theorem of threeway concept lattice. Int J Approx Reason 146:157–173. https://doi.org/10.1016/j.ijar.2022.04.007
Gao N, Cao Z, Li Q, Yao W, Jiang H (2022) Latticetheoretic threeway formal contexts and their concepts. Int J Approx Reason 26:8971–8985. https://doi.org/10.1007/s00500022072943
Qi J, Qian T, Wei L (2016) The connections between threeway and classical concept lattices. KnowlBased Syst 91(JAN.):143–151. https://doi.org/10.1016/j.knosys.2015.08.006
Qian T, Wei L, Qi J (2017) Constructing threeway concept lattices based on apposition and subposition of formal contexts. KnowlBased Syst 116(15):39–48. https://doi.org/10.1016/j.knosys.2016.10.033
Yang SC, Lu YN, Jia XY, Li WW (2020) Constructing threeway concept lattice based on the composite of classical lattices. Int J Approx Reason 121:174–186. https://doi.org/10.1016/j.ijar.2020.03.007
Wei L, Liu L, Qi J, Qian T (2019) Rules acquisition of formal decision contexts based on threeway concept lattices. Inf Sci 516:529–544. https://doi.org/10.1016/j.ins.2019.12.024
Burusco A, FuentesGonzález R (1998) Construction of the lfuzzy concept lattice. Fuzzy Sets Syst 97(1):109–114. https://doi.org/10.1016/s01650114(96)003181
Duntsch I, Gediga G (2002) Modalstyle operators in qualitative data analysis. In: Proceedings of the 2002 IEEE international conference on data mining (ICDM 2002), 912 December 2002, Maebashi City, Japan, pp 155–162. https://doi.org/10.1109/ICDM.2002.1183898
Yao YY (2004) Concept lattices in rough set theory. In: IEEE Annual meeting of the fuzzy information, 2004. Processing NAFIPS’04, vol 2, pp 796–801. https://doi.org/10.1109/NAFIPS.2004.1337404
Li JH, Mei CL, Lv Y (2013) Incomplete decision contexts: approximate concept construction, rule acquisition and knowledge reduction  sciencedirect. Int J Approx Reason 54(1):149–165. https://doi.org/10.1016/j.ijar.2012.07.005
Fan B, Tsang ECC, Xu W, Chen D, Li W (2019) Attributeoriented cognitive concept learning strategy: a multilevel method. Int J Mach Learn Cybern 10:2421–2437. https://doi.org/10.1007/s1304201808795
Li JH, Ren Y, Mei CL, Qian YH, Yang XB (2016) A comparative study of multigranulation rough sets and concept lattices via rule acquisition. KnowlBased Syst 91:152–164. https://doi.org/10.1016/j.knosys.2015.07.024
Shao MW, Lv MM, Li KW, Wang CZ (2019) The construction of attribute (object)oriented multigranularity concept lattices. Int J Mach Learn Cybern 1–16. https://doi.org/10.1007/s13042019009550
Zhi H, Li J (2019) Granule description based knowledge discovery from incomplete formal contexts via necessary attribute analysis. Inf Sci 485:347–361. https://doi.org/10.1016/j.ins.2019.02.032
Qian T, Wei L, Qi J (2019) A theoretical study on the object (property) oriented concept lattices based on threeway decisions. Soft Comput 23:9477–9489. https://doi.org/10.1007/s00500019037996
Zhi HL, Qi JJ (2019) Qian T, Wei L: Threeway dual concept analysis. Int J Approx Reason 114:151–165. https://doi.org/10.1016/j.ijar.2019.08.010
Akram M, Ali G, Alcantud JCR (2021) Parameter reduction analysis under intervalvalued mpolar fuzzy soft information. Artif Intell Rev 54(7):5541–5582. https://doi.org/10.1007/s1046202110027x
Akram M, Ali G, Alcantud JCR (2022) Attributes reduction algorithms for mpolar fuzzy relation decision systems. Int J Approx Reason 140:232–254. https://doi.org/10.1016/j.ijar.2021.10.005
Akram M, Ali G, Alcantud JC, Fatimah F (2021) Parameter reductions in nsoft sets and their applications in decisionmaking. Exp Syst 38(1):12601. https://doi.org/10.1111/exsy.12601
Huang J, Lin Y, Li J (2023) Rule reductions of decision formal context based on mixed information. Appl Intell 53(12):15459–15475. https://doi.org/10.1007/s10489022041949
Aragón R, Medina J, RamírezPoussa E (2021) Reducing concept lattices by means of a weaker notion of congruence. Fuzzy Sets Syst 418(1):153–169. https://doi.org/10.1016/j.fss.2020.09.013
Aragón R, Medina J, RamírezPoussa E (2021) Identifying nonsublattice equivalence classes induced by an attribute reduction in fca. Mathematics 6(5):2–15. https://doi.org/10.3390/math9050565
Zhao S, Qi J, Li J, Wei L (2023) Concept reduction in formal concept analysis based on representative concept matrix. Int J Mach Learn Cybern 14(4):1147–1160. https://doi.org/10.1007/s13042022016918
Wu WZ, Leung Y, Mi JS (2009) Granular computing and knowledge reduction in formal contexts. IEEE Trans Knowl Data Eng 21(10):1461–1474. https://doi.org/10.1109/TKDE.2008.223
Li J, Kumar CA, Mei C, Wang X (2017) Comparison of reduction in formal decision contexts. Int J Approx Reason 80(Jan.):100–122. https://doi.org/10.1016/j.ijar.2016.08.007
Qin K, Li B, Pei Z (2019) Attribute reduction and rule acquisition of formal decision context based on object (property) oriented concept lattices. Int J Mach Learn Cybern 10:2837–2850. https://doi.org/10.1007/s13042018009070
Li J, Mei C, Wang J, Xiao Z (2014) Rulepreserved object compression in formal decision contexts using concept lattices. KnowlBased Syst 71(nov.):435–445. https://doi.org/10.1016/j.knosys.2014.08.020
Shi LL, Yang HL (2018) Object granular reduction of fuzzy formal contexts. J Intell Fuzzy Syst 1(34):633–644. https://doi.org/10.3233/JIFS17909
Ren R, Ling W (2016) The attribute reductions of threeway concept lattices. KnowlBased Syst 99:92–102. https://doi.org/10.1016/j.knosys.2016.01.045
Shao MW, Yang HZ, Wu WZ (2015) Knowledge reduction in formal fuzzy contexts. KnowlBased Syst 73(jan.):265–275. https://doi.org/10.1016/j.knosys.2014.10.008
Wang Z, Shi C, Wei L, Yao Y (2023) Trigranularity attribute reduction of threeway concept lattices. KnowlBased Syst 276:1–14. https://doi.org/10.1016/j.knosys.2023.110762
Shao MW, Li KW (2016) Attribute reduction in generalized onesided formal contexts. Inf Sci 378:317–327. https://doi.org/10.1016/j.ins.2016.03.018
Lin Y, Li J, Tan A, Zhang J (2020) Granular matrixbased knowledge reductions of formal fuzzy contexts. Int J Mach Learn Cybern 11(3):643–656. https://doi.org/10.1007/s13042019010224
Zhang C, Li J, Lin Y (2021) Matrixbased reduction approach for onesided fuzzy threeway concept lattices. J Intell Fuzzy Syst 40(6):11393–11410. https://doi.org/10.3233/JIFS202573
Janostik R, Konecny J (2020) General framework for consistencies in decision contexts. Inf Sci 530:180–200. https://doi.org/10.1016/j.ins.2020.02.045
Li JY, Wang X, Wu WZ, Xu YH (2017) Attribute reduction in inconsistent formal decision contexts based on congruence relations. Int J Mach Learn Cybern 8:81–94. https://doi.org/10.1007/s130420160586z
Herrera F, Martinez L (2000) A 2tuple fuzzy linguistic representation model for computing with words. IEEE Trans Fuzzy Syst 8(6):746–752. https://doi.org/10.1109/91.890332
Xu Z (2004) A method based on linguistic aggregation operators for group decision making with linguistic preference relations. Inf Sci 166(1–4):19–30. https://doi.org/10.1016/j.ins.2003.10.006
Rodriguez RM (2012) Martinez L, Herrera F: Hesitant fuzzy linguistic term sets for decision making. IEEE Trans Fuzzy Syst 20(1):109–119. https://doi.org/10.1109/TFUZZ.2011.2170076
Pang Q, Wang H, Xu Z (2016) Probabilistic linguistic term sets in multiattribute group decision making. Inf Sci 128–143. https://doi.org/10.1016/j.ins.2016.06.021
Ji F, Wu J, Chiclana F, Wang S, Fujita H, HerreraViedma E (2023) the overlapping community driven feedback mechanism to support consensus in social network group decision making. IEEE Trans Fuzzy Syst 31(9):3025–3039. https://doi.org/10.1109/TFUZZ.2023.3241062
Sun Q, Chiclana F, Wu J, Liu Y, Liang C, HerreraViedma E (2023) Weight penalty mechanism for noncooperative behavior in largescale group decision making with unbalanced linguistic term sets. IEEE Trans Fuzzy Syst 31(10):3507–3521. https://doi.org/10.1109/TFUZZ.2023.3260820
Zou L, Pang K, Song X, Kang N, Liu X (2020) A knowledge reduction approach for linguistic concept formal context  sciencedirect. Inf Sci 524:165–183. https://doi.org/10.1016/j.ins.2020.03.002
Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoningi. Inf Sci 8(3):199–249. https://doi.org/10.1016/00200255(75)900365
Herrera F, Herrera V, Verdegay J (1996) A model of monsensus in group decision making under linguistic assessments. Fuzzy Sets Syst 78(1):73–87. https://doi.org/10.1016/01650114(95)001077
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The authors thank reviewers for their valuable feedback, which led to significant improvements of the manuscript.
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This work was supported by the National Natural Science Foundation of P.R. China (Nos.62176142), Foundation of Liaoning Educational Committee (No.LJ2020007) and Special Foundation for Distinguished Professors of Shandong Jianzhu University.
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Investigation, H.C.(Hui Cui) and L.Z.(Li Zou); writingoriginal draft preparation, H.C.; writingreview and editing, H.C. and T.H.(Tie Hou); supervision, L.Z. and A.D. (Ansheng Deng); formal analysis, L.M.(Luis Martine); project administration, L.Z. and A.D. All authors have read and agreed to the published version of the manuscript.
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Cui, H., Deng, A., Hou, T. et al. Exploring object reduction approaches for optimizing decisionmaking in linguistic concept formal context. Appl Intell 54, 9088–9104 (2024). https://doi.org/10.1007/s1048902405583y
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DOI: https://doi.org/10.1007/s1048902405583y