Exploring object reduction approaches for optimizing decision-making in linguistic concept formal context

Abstract

Knowledge reduction is a crucial research topic in formal concept analysis. Given the ability of three-way 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 three-way concept lattice. We achieve this by integrating necessity and possibility operators in a formal context containing linguistic data. Our proposed approach involves constructing an attribute-induced three-way object-oriented linguistic concept lattice to classify the object set into three regions based on three-way decision. Additionally, we introduce an approach of attribute-induced three-way object-oriented 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 attribute-induced three-way object-oriented 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 decision-making challenges in corporate projects using different attribute-induced three-way object-oriented linguistic concepts.

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].

Three-way 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 three-way concept lattice [8] integrates the idea of three-way decisions into formal concept analysis, providing a more comprehensive understanding of decision-making problems. The three-way concept lattice not only reflects the specific expression and significance of FCA but also embodies the idea of three divisions in three-way decisions, which recognizes the importance of uncertainty in decision-making and provides a framework for evaluating and managing it. By utilizing the three-way concept lattice, decision-makers 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 three-way concept lattice method by constructing a new isomorphic formal context that combines the given formal context with its complementary context. The object-induced three-way operators and attribute-induced three-way operators have been introduced in [14], and the corresponding candidate and redundant concepts have been presented to build the three-way concept lattice simply and effectively. Considering that the three-way concept lattice can express both positive and negative information in a formal context, a rule extraction method of the three-way concept lattice has been discussed [15].

Besides, various concepts have been proposed to describe different objective things, such as fuzzy concepts [16], attribute-oriented concepts [17], object-oriented concepts [18], approximate concepts [19] and so on. Among them, the object-oriented concept lattice and the attribute-oriented 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 large-scale 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 multi-granularity 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 three-way decisions, the relations between object-oriented concept lattice, attribute-oriented concept lattice, three-way object-oriented concept lattice, and three-way attribute-oriented 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 three-way dual concept lattice and analyzed four models based on three-way 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 attribute-induced three-way concept lattice and object-induced three-way 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 three-way concept lattices. By incorporating new levels with existing global granularity reduction, they have established a framework for tri-granularity attribute reduction in three-way concept lattices. In [41], the lattice structure in generalized one-sided 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 crisp-fuzzy concepts in fuzzy formal context, and then presented a granular matrix-based 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 decision-making problems in uncertain environments. Taking full advantage of linguistic term sets, multiple uncertainty linguistic representation models have been proposed, such as 2-tuple 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 decision-making 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 (SN-GDM) and proposed an overlapping community-driven feedback mechanism to improve consensus. Sun et al. [51] have presented a novel framework for addressing noncooperative behavior within subgroups in large-scale group decision-making processes. By minimizing adjustment costs and optimizing penalty parameters, this method effectively prevents excessive penalization, thereby enhancing the efficiency of collaborative decision-making 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 three-way concept lattices has primarily concentrated on the presence or absence of attributes in objects, without considering the broader object-attribute relationships. For instance, in a decision-making 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 decision-making 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 NP-hard problem. This complexity amplifies the difficulty of employing concept lattice methodologies for knowledge extraction, demanding careful optimization to mitigate resource-intensive 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:

• Attribute-induced three-way object-oriented linguistic concept lattice: the proposal of an innovative attribute-induced three-way object-oriented linguistic concept lattice model that integrates possibility and necessity operators into the linguistic concept formal context to describe three-way 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, meet-irreducible element, join-irreducible element, and equivalence relations.

• Decision-making 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 decision-making optimization.

The rest of this paper is organized as follows. Section 2 reviews some basic notions on concept lattice, three-way concept lattice, linguistic term set and linguistic concept lattice. In Section 3, we first construct an attribute-induced three-way object-oriented 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:

$$\begin{aligned} X^{\prime }=\{a\in {M}|\forall {x\in {X}},(x,a)\in {I}\}, \end{aligned}$$

(1)

$$\begin{aligned} B^{*}=\{x\in {G}|\forall {a\in {B}},(x,a)\in {I}\}. \end{aligned}$$

(2)

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:

$$\begin{aligned} (X_{1}, B_{1})\le {(X_{2}, B_{2})}\Leftrightarrow {X_{1}\subseteq {X_{2}}(\Leftrightarrow {B_{2}\subseteq {B_{1}}}}), \\ (X_{1}, B_{1})\wedge {(X_{2}, B_{2})}=(X_{1}\cap {X_{2}},(B_{1}\cup {B_{2}})^{*\prime }),\end{aligned}$$

(3)

$$\begin{aligned} (X_{1}, B_{1})\vee {(X_{2}, B_{2})}=((X_{1}\cup {X_{2}})^{\prime *},B_{1}\cap {B_{2}}). \end{aligned}$$

(4)

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,

$$\begin{aligned} X^{\overline{\prime }}=\{a\in {M}|\forall {x\in {X}},\lnot (xIa)\}=\{a\in {M}|\forall {x\in {X}, (x,a)\in {I^{c}}}\},\end{aligned}$$

(5)

$$\begin{aligned} B^{\overline{*}}=\{x\in {G}|\forall {a\in {B}},\lnot (xIa)\}=\{x\in {G}|\forall {a\in {B}, (x,a)\in {I^{c}}}\}. \end{aligned}$$

(6)

Here, $I^{c}=(G\times {M})-I$. If $X^{\overline{\prime }}=B$ and $B^{\overline{*}}=X$, then we call (X, B) an N-concept.

The notation NL(G, M, I) represents a complete lattice consisting of all the N-concepts. 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 Attribute-induced three-way concept lattice

Inspired by the idea of three-way decision, Qi et al. [8] have put forward three-way concept lattices in terms of both objects and attributes, where the attribute-induced three-way 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 attribute-induced three-way 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 attribute-induced three-way concept, or an AE-concept 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 AE-concept ((X, Y), B).

The AE-concepts $((X_{1},Y_{1}),B_{1})$ and $((X_{2},Y_{2}),B_{2})$ are ordered by:

$$\begin{aligned} ((X_{1},Y_{1}),B_{1})\le {((X_{2},Y_{2}),B_{2})}\Leftrightarrow {(X_{1},Y_{1})\subseteq {(X_{2},Y_{2})}}(\Leftrightarrow {B_{1}\supseteq {B_{2}}})\nonumber . \end{aligned}$$

(7)

Further, all AE-concepts form a complete lattice which called an attribute-induced three-way concept lattice(AE-lattice), 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 non-empty finite object set, $L_{S} = \{l^{j}_{s_{\alpha }}|j={1,2,3,...,m}, \alpha ={0, 1, 2,. . ., g}\}$ is a non-empty 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:

$$\begin{aligned} X^{\prime }=\{l_{s}\in {L_{S}}|\forall {x\in {X}},(x, l_{s})\in {\mathfrak {I}}\}\ , \end{aligned}$$

(8)

$$\begin{aligned} B_{S}^{*}=\{x\in {G}|\forall {l_{s}\in {B_{S}}},(x, l_{s})\in {\mathfrak {I}}\}. \end{aligned}$$

(9)

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:

$$\begin{aligned} (X_{1}, B^{1}_{S})\le (X_{2}, B^{2}_{S})\Leftrightarrow X_{1}\subseteq {X_{2}}(\Leftrightarrow B^{2}_{S}\subseteq B^{1}_{S}),\\ (X_{1}, B^{1}_{S})\wedge {(X_{2}, B^{2}_{S})}=(X_{1}\cap {X_{2}},(B^{1}_{S}\cup {B^{2}_{S}})^{*\prime }),\end{aligned}$$

(10)

$$\begin{aligned} (X_{1}, B^{1}_{S})\vee {(X_{2}, B^{2}_{S})}=((X_{1}\cup {X_{2}})^{\prime *},B^{1}_{S}\cap {B^{2}_{S}}). \end{aligned}$$

(11)

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, a-leadership skill, b-professional knowledge skill and c-communication 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.

Table 1 Linguistic concept formal context $(G, L_{S}, \mathfrak {I})$

Fig. 1

Linguistic concept lattice $LL(G, L_{S}, \mathfrak {I})$

3 Object reductions of attribute-induced three-way object-oriented 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 decision-making problems that involve attribute values expressed in linguistic terms. To address this, we study the three-way linguistic concept lattice structure and reduction approaches which are generated by modal-style approximate operators in a linguistic concept formal context.

3.1 The construction of attribute-induced three-way object-oriented 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:

$$\begin{aligned} x^{\prime }=\{l_{s}\in {L_{S}}|\forall {x\in {G}},(x, l_{s})\in {\mathfrak {I}}\}, \end{aligned}$$

(12)

$$\begin{aligned} l_{s}^{*}=\{x\in {G}|\forall {l_{s}\in {L_{S}}},(x, l_{s})\in {\mathfrak {I}}\},\end{aligned}$$

(13)

$$\begin{aligned} X^{\Box }=\{l_{s}\in {L_{S}}|l_{s}^{*}\subseteq {X}\},\end{aligned}$$

(14)

$$\begin{aligned} B_{S}^{\diamond }=\{x\in {G}|x^{\prime }\cap {B_{S}}\ne {\emptyset }\}. \end{aligned}$$

(15)

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:

$$\begin{aligned} x^{\overline{\prime }}=\{l_{s}\in {L_{S}}|\forall {x\in {G}},(x, l_{s})\in {\mathfrak {I}^{c}}\}, \end{aligned}$$

(16)

$$\begin{aligned} l_{s}^{\overline{*}}=\{x\in {G}|\forall {l_{s}\in {L_{S}},(x, l_{s})\in {\mathfrak {I}^{c}}}\},\end{aligned}$$

(17)

$$\begin{aligned} X^{\overline{\Box }}=\{l_{s}\in {L_{S}}|l_{s}^{\overline{*}}\subseteq {X}\},\end{aligned}$$

(18)

$$\begin{aligned} B_{S}^{\overline{\diamond }}=\{x\in {G}|x^{\overline{\prime }}\cap {B_{S}}\ne {\emptyset }\}. \end{aligned}$$

(19)

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 attribute-induced three-way object-oriented linguistic operators (AEOL-operators) 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 attribute-induced three-way object-oriented linguistic concept, for short, an AEOL-concept 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 AEOL-concept.

For two AEOL-concepts $((X_{1}, Y_{1}),B^{1}_{S})$ and $((X_{2}, Y_{2}),$ $B^{2}_{S}),$ the partial order relation between them is defined as

$$\begin{aligned} ((X_{1}, Y_{1}),B^{1}_{S}) \le { ((X_{2}, Y_{2}),B^{2}_{S})}\Leftrightarrow {(X_{1}, Y_{1})\subseteq {(X_{2}, Y_{2})}}\Leftrightarrow {B^{1}_{S}}\subseteq {B^{2}_{S}},\nonumber \end{aligned}$$

(20)

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 sub-concept of $((X_{2}, Y_{2}),B^{2}_{S})$, and $((X_{2}, Y_{2}),B^{2}_{S})$ is called a super-concept of $((X_{1}, Y_{1}),B^{1}_{S})$.

For linguistic concept subset $B_{S}\subseteq {L_{S}}$, the AEOL-operators can divide the object set G into three regions, $POS_{B_{S}}=G-B_{S}^{\overline{\diamond }}$, $NEG_{B_{S}}=G-B_{S}^{\diamond }$ 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 AEOL-concepts can express the meaning of the attribute-induced three-way concepts.

Similarly, we can obtain the properties corresponding to attribute-induced three-way object-oriented 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 AEOL-concepts is called an attribute-induced three-way object-oriented linguistic concept lattice of $(G, L_{S}, \mathfrak {I})$, which is a complete lattice. Given two AEOL-concepts $((X_{1}, Y_{1}),B^{1}_{S})$ and $((X_{2}, Y_{2}),B^{2}_{S})$, the supremum and infimum are defined as follows:

$$\begin{aligned} ((X_{1}, Y_{1}),B^{1}_{S})\vee { ((X_{2}, Y_{2}),B^{2}_{S})}=(((X_{1}, Y_{1})\cup {(X_{2}, Y_{2})})^{{\trianglelefteq }{\trianglerighteq }},B^{1}_{S}\cup {B^{2}_{S}}), \nonumber \\ \end{aligned}$$

(21)

$$\begin{aligned} ((X_{1}, Y_{1}),B^{1}_{S})\wedge { ((X_{2}, Y_{2}),B^{2}_{S})}=((X_{1}, Y_{1})\cap {(X_{2}, Y_{2})},(B^{1}_{S}\cap {B^{2}_{S}})^{{\trianglerighteq }{\trianglelefteq }}).\nonumber \\ \end{aligned}$$

(22)

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 decision-making 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 attribute-induced three-way object-oriented 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 AEOL-concepts 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 AEOL-concept 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 AEOL-concept $((x_{3}x_{4}x_{5},x_{1}x_{2}x_{5}), b_{s_{2}}c_{s_{1}})$ 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, AEOL-concepts 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 non-possessed between objects and linguistic concepts but also captures the complement relations of groups formed by these objects.

Table 2 AEOL-concepts of Table 1

Fig. 2

AEOL lattice of $(G, L_{S}, \mathfrak {I})$

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 AEOL-concepts.

Definition 12

Let $(G, L_{S}, \mathfrak {I})$ be a linguistic concept formal context, $\forall l_{s}\in {L_{S}}$, the granular concept of AEOL-concept 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 attribute-induced three-way object-oriented 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

$$\begin{aligned} DIS_{AEOLG}(p,q)=(p^{\diamond }-q^{\diamond }, p^{\overline{\diamond }}-q^{\overline{\diamond }}), \end{aligned}$$

(23)

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 AEOLG-discernibility matrix, where $\Lambda$ represents the non-empty object set of AEOLG-discernibility matrix, then we propose the AEOLG-discernibility function.

Definition 15

Let $(G, L_{S}, \mathfrak {I})$ be a linguistic concept formal context, the AEOLG-discernibility function is defined as follows.

$$\begin{aligned} f(\Lambda _{AEOLG})=\bigwedge \bigvee \limits _{p,q\in {L_{S}}}((p^{\diamond }-q^{\diamond })\bigcup ({p^{\overline{\diamond }}-q^{\overline{\diamond }}})), \end{aligned}$$

(24)

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 $\frac{t^{2}-2t+5}{4}t^{2}|G||L_{S}|^{2}$.

Algorithm 1

Granular reduction algorithm in linguistic concept formal context $(G, L_{S}, \mathfrak {I})$.

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 AEOL-concepts 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 AEOLG-discernibility matrix is shown in Table 3, the AEOLG-discernibility 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 AEOL-concepts 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 AEOL-concept 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 AEOL-concepts 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 AEOL-concepts shown in Example 2, regardless of whether the employee is $x_{3}$ or $x_{4}$. The granular concepts of AEOL-concepts 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 AEOL-concepts, 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 $AEOLL(G, L_{S}, \mathfrak {I})$.

Table 3 AEOLG-discernibility matrix

Table 4 AEOL-concepts under object granular reduction $\{x_{2}, x_{3},$ $x_{5}\}$

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 AEOL-concepts 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 attribute-induced three-way object-oriented 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 meet-irreducible elements of $AEOLL(G, L_{S}, \mathfrak {I})$ and denote $Int_{AEOLJ}(G,$ $L_{S}, \mathfrak {I})$ as the intent set of all the join-irreducible 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 attribute-induced three-way object-oriented linguistic meet-irreducible element (AEOL-MIE) 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 AEOL-MIE preserving consistent set, then Z is called an AEOL-MIE preserving reduction.

Theorem 8

Let $(G, L_{S}, \mathfrak {I})$ be a linguistic concept formal context, $Z\subseteq {G}$. Then, Z is an AEOL-MIE 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 AEOL-MIE 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 AEOL-meet-irreducible 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 AEOL-meet-irreducible 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 AEOL-MIE 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 AEOL-MIE 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 attribute-induced three-way object-oriented linguistic join-irreducible element (AEOL-JIE) 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 AEOL-JIE preserving consistent set, then Z is called an AEOL-JIE 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 attribute-induced three-way object-oriented linguistic join-irreducible element (AEOL-JIE) if and only if $((X, Y), B_{S})$ is the granular concept of AEOL-concepts.

Proof

$(\Rightarrow )$ Assume that $((X, Y), B_{S})$ is an attribute-induced three-way object-oriented linguistic join-irreducible element, then for any intent $B^{j}_{S}\in {L_{S}}$ of AEOL-JIE, 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 AEOL-concept, 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 AEOL-concepts.

$(\Leftarrow )$ 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 AEOL-JIE 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 AEOL-JIE 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:

$$\begin{aligned} R_{Z}^{\diamond }=\{(p,q)\in {L_{S}\times {L_{S}}}|p^{\diamond }_{Z}=q^{\diamond }_{Z}\,\,\, or \,\, p^{\overline{\diamond }}_{Z}=q^{\overline{\diamond }}_{Z}\}.\ \end{aligned}$$

(25)

If $R_{Z}^{\diamond }=R_{G}^{\diamond }$, then Z is referred to as an attribute-induced three-way object-oriented 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 meet-irreducible element (AEOL-MIE) 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

$$\begin{aligned} DIS_{AEOLL}(((X, Y), B_{S}),((Z, W), C_{S}))=\\ \nonumber \left\{ \begin{array}{l} (X-Z, Y-W), if ((X, Y), B_{S})\prec {((Z, W), C_{S})}, \\ \emptyset , \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad otherwise. \\ \end{array} \right. \end{aligned}$$

(26)

Where $DIS_{AEOLL}(((X, Y), B_{S}),((Z, W), C_{S}))$ is called the AEOLL-discernibility 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 AEOLL-discernibility matrix, then the AEOLL-discernibility function is proposed.

Definition 21

Let $(G, L_{S}, \mathfrak {I})$ be a linguistic concept formal context, the AEOLL-discernibility function is defined as follows.

$$\begin{aligned} f(\Lambda _{AEOLL})=f(\Lambda _{AEOLM})=\bigwedge \limits _{H\in {\Lambda _{AEOLL}}}(\bigvee \limits _{h\in {H}}h). \end{aligned}$$

(27)

Table 5 AEOL-concepts under AEOLL reductions $\{x_{1}, x_{2}, x_{3}, x_{5}\}$

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.

Algorithm 2

Lattice reduction algorithm in linguistic concept formal context $(G, L_{S}, \mathfrak {I})$.

Example 4

Considering the linguistic concept formal context $(G, L_{S}, \mathfrak {I})$ in Table 1, we can obtain the AEOLL reductions and AEOL-MIE 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 AEOL-JIE 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 AEOL-concepts 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 AEOL-concepts obtained from the three methods in Tables 2, 4, and 5:

AEOL-concept based on Table 2: $((x_{1}x_{2}x_{5}, x_{2}x_{3}x_{4}x_{5}), a_{s_{2}}$ $b_{s_{1}}c_{s_{2}})$.

AEOL-concept based on Table 4: $((x_{2}x_{5},x_{2}x_{3}x_{5}), a_{s_{1}}a_{s_{2}}$ $b_{s_{0}}b_{s_{1}}c_{s_{0}}c_{s_{2}})$.

AEOL-concept 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 decision-making 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.

Fig. 3

The optimal decisions of above examples

Fig. 4

The relations among the five consistent sets

The examples provided above regarding the optimization decision-making 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 AEOL-concepts 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 AEOL-concepts 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 AEOL-MIE, AEOL-JIE, AEOLG, and AEOLC. Furthermore, AEOLL reduction is equivalent to AEOL-MIE reduction; hence, the necessary information for AEOLL reduction can be obtained by computing AEOL-MIE reduction. Additionally, it can be observed that AEOLG reduction is equivalent to reductions of AEOLC and AEOL-JIE. Therefore, maintaining join-irreducible 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.

Table 6 The comparison between reduction methods

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 two-dimensional 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 real-life 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. AEOL-MIE and AEOL-JIE 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 three-way decision theory is widely recognized as a practical approach for solving decision-making 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 decision-making problems with linguistic data. Specifically, we have introduced a granular reduction approach that combines three-way 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 decision-making problems of different queries in project assignment within a company, the effective identification of the optimal employee or team corresponding to the AEOL-concepts can be achieved by employing different object reduction methods, which emphasizes the distinct advantages of these object reduction methods in decision-making 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 decision-making 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.