An Order-Theoretic Study on Formal Concept Analysis

Abstract

This paper employs an order-theoretic framework to explore the intricacies of formal concepts. Initially, we establish a natural correspondence among formal contexts, preorders, and the resulting partially ordered sets (posets). Leveraging this foundation, we provide insightful characterizations of atoms and coatoms within finite concept lattices, drawing upon object intents. Expanding from the induced poset originating from a formal context, we extend these characterizations to discern join-irreducible and meet-irreducible elements within finite concept lattices. Contrary to a longstanding misunderstanding, our analysis reveals that not all object and attribute concepts are irreducible. This revelation challenges the conventional belief that rough approximations, grounded in irreducible concepts, offer sufficient coverage. Motivated by this realization, the paper introduces a novel concept: rough conceptual approximations. Unlike the conventional definition of object equivalence classes in Pawlakian approximation spaces, we redefine them by tapping into the extent of an object concept. Demonstrating their equivalence, we establish that rough conceptual approximations align seamlessly with approximation operators in the generalized approximation space associated with the preorder corresponding to a formal context. To illustrate the practical implications of these theoretical findings, we present concrete examples. Furthermore, we delve into the significance and potential applications of our proposed rough conceptual approximations, shedding light on their utility in real-world scenarios.

1. Introduction

Formal concept analysis (FCA) and rough set theory (RST), introduced by Rudolf Wille [1] and Zdzislaw Pawlak [2], respectively, stand as two prominent mathematical theories meticulously crafted to enhance the landscape of data analysis and knowledge processing. These theoretical frameworks, designed with precision and purpose, have evolved into indispensable tools for knowledge extraction and acquisition from complex datasets. Their proven effectiveness underscores their pivotal role in navigating the intricacies of data, establishing them as powerful instruments for advancing the field of knowledge processing.

FCA is initiated by a formal context (or simply context) $(G,M,I)$, where G and M are nonempty sets comprising objects and attributes, respectively, and $I\subseteq G\times M$ is a binary incidence relation connecting objects and attributes. Specifically, $(g,m)\in I$ indicates that object g has attribute m.

Given a formal context $(G,M,I)$, we have two basic operators, ${(·)}^{\uparrow }:\wp (G)⟶\wp (M)$ and ${(·)}^{\downarrow }:\wp (M)⟶\wp (G)$ defined by

$$A^{\uparrow }=\{m\in M\mid \forall g\in A,(g,m)\in I\},\mathrm{and}$$

$$B^{\downarrow }=\{g\in G\mid \forall m\in B,(g,m)\in I\},$$

for any $A\subseteq G$ and $B\subseteq M$, where $\wp (X)$ denotes the powerset of a set X. Intuitively, $A^{\uparrow }$ is the maximal set of attributes shared by all objects in A, and $B^{\downarrow }$ is the maximal set of objects that possess all attributes in B.

In a context, a formal concept is a pair $(A,B)\in \wp (G)\times \wp (M)$ such that

$$A^{\uparrow }=B\mathrm{and}{B}^{\downarrow }=A,$$

(1)

where A and B are called the extent and the intent of the formal concept, respectively. For a given $g\in G$ and $m\in M$, ${\left\{g\right\}}^{\uparrow }$ and ${\left\{m\right\}}^{\downarrow }$ are called the object intent of g and the attribute extent of m, respectively. The concept $({({\left\{g\right\}}^{\uparrow })}^{\downarrow }$, ${\left\{g\right\}}^{\uparrow })$ (respectively, $({\left\{m\right\}}^{\downarrow },{({\left\{m\right\}}^{\downarrow })}^{\uparrow })$) is the object (respectively, attribute) concept associated with g (respectively, m).

The formal concepts of $(G,M,I)$ are partially ordered based on the inclusion between extents or, equivalently, the inverse inclusion between intents. Formally,

$$(A_1,B_1)\le (A_2,B_2):⟺A_1\subseteq A_2(⟺B_1\supseteq B_2).$$

(2)

Equipped with this partial order ≤, the set $\mathcal{B}(G,M,I)$ consisting of all formal concepts of $(G,M,I)$ forms a complete lattice (a partially ordered set in which every subset has an infimum and a supremum), referred to as the concept lattice of $(G,M,I)$ [3].

The concept of formal contexts serves as an integrative framework that unifies both RST and FCA [4]. At the core of rough set philosophy is the foundational idea of treating objects as indiscernible when they share identical attributes. This philosophical stance translates into a mathematical representation through the indiscernibility relation, establishing an equivalence relation that identifies all indiscernible objects. For a set X of objects, the lower approximation of X encompasses objects whose equivalence classes are subsets of X. Conversely, the upper approximation of X includes objects whose equivalence classes intersect nontrivially with X. This nuanced approach captures the essence of both RST and FCA, highlighting the intricate interplay between discernibility and equivalence within the broader framework of formal contexts.

In the application of RST to formal contexts, Kent [5] delves into the exploration of rough approximations within conceptual structures. Yet, as underscored by Saquer and Deogun [6], Kent’s methodology hinges on the availability of an expert-provided equivalence relation for the set of objects. This dependency introduces a significant caveat, as the outcomes vary depending on the specific equivalence relation chosen. In essence, the robustness and reliability of Kent’s approach are contingent upon the subjectivity inherent in the selection of an equivalence relation, thereby warranting a careful consideration of its impact on the results.

In response to this limitation, Saquer and Deogun [6] devised a methodology for approximating concepts within the FCA framework. Operating within a given formal context $(G,M,I)$, they introduced an equivalence relation on G, defining two objects as equivalent if they share the same set of attributes. This innovative approach offered a fresh perspective on concept approximation, seamlessly integrating RST into the FCA framework. Crucially, their proposed equivalence relation aligns precisely with the indiscernibility relation inherent in RST, establishing a harmonious bridge between the two theories. This strategic alignment not only overcomes the limitations associated with expert-dependent equivalence relations but also enhances the robustness and consistency of the concept approximation process within the broader context of FCA.

Nonetheless, as highlighted in [7], it is crucial to recognize that an equivalence class of objects may not necessarily align with any extent of formal concepts. Consequently, employing equivalence classes as definable sets within the context of FCA is deemed unreasonable. In response to this challenge, the suggested approach is to utilize join-irreducible or meet-irreducible concepts as the foundational building blocks for constructing rough approximations within formal contexts. This strategic shift seeks to overcome the inherent limitations associated with equivalence classes, offering a more rational and effective means of delineating rough approximations in the realm of FCA.

As highlighted in [3], each formal concept (A, B) within a context (G, M, I) serves as the supremum (or infimum) of certain object (or attribute) concept. Contrary to conventional wisdom, this paper demonstrates that in a finite concept lattice, not all object (or attribute) concepts are necessarily join-irreducible (or meet-irreducible), as elucidated in our findings. This revelation challenges the completeness of the basic building blocks proposed in [7]. Recognizing this gap, we present a novel solution to the issue by introducing a distinctive definition of rough approximation within formal contexts. In our approach, we consider object (or attribute) concepts as the fundamental building blocks, providing a more comprehensive and refined perspective that addresses the limitations of existing frameworks. This innovative definition enhances the robustness and completeness of rough approximations within formal contexts, offering a contribution to the ongoing discourse in the field.

The paper is organized as follows. In the subsequent part of this section, we illuminate the principal contributions made by this work and draw comparisons with existing literature. Section 2 furnishes essential background information encompassing preorders; equivalence relations; posets; lattices; FCA, including the foundational basic theorem on concept lattices; and RST. Moving to Section 3, we initially establish characteristic conditions elucidating the irreducibility of object and attribute concepts. This exploration reveals that irreducible concepts, despite their prominence, exhibit limitations in providing sufficient coverage for rough approximations. Consequently, in this section, we introduce a groundbreaking definition of rough conceptual approximations designed to surmount this coverage challenge. Section 4 navigates through the practical implications of our findings by illustrating them with examples. Additionally, we present a comparative analysis and delve into potential applications, underscoring the significance and versatility of our contribution. Lastly, Section 5 concludes with insightful remarks that encapsulate the key takeaways and implications of our work.

1.1. Main Contributions

Given that the approximation space serves as the abstract structure for information systems in RST for intelligent data analysis, a pertinent question arises regarding the corresponding abstract structure for formal contexts in FCA; while one could argue that, trivially, the approximation space aligns with the abstract structure since a formal context is a specialized information system, this paper offers a more nuanced perspective. In our exploration, we propose a less obvious answer to this question, contending that the abstract structure for formal contexts is, precisely, the preordered set. We establish a direct association between each formal context and a natural preordered set, concurrently establishing a canonical formal context for each preordered set. This order-theoretic investigation yields several theoretical findings and potential applications.

• We introduce a novel concept, termed rough conceptual approximations, demonstrating its equivalence to approximation operators within the preordered generalized approximation space.
• We present a comprehensive characterization of join-irreducible and meet-irreducible elements within finite concept lattices. This revelation challenges the completeness of rough approximations founded on irreducible concepts, as proposed in [7], motivating the introduction of our novel definition of rough conceptual approximations.

In essence, our contributions transcend the conventional understanding of the abstract structures associated with formal contexts, offering a more refined and comprehensive theoretical framework for intelligent data analysis within the FCA domain.

1.2. Related Work

The interaction between FCA and RST has been studied extensively. At the beginning, Kent introduced rough concept analysis by equipping the formal context with an expert-provided equivalence relation over the set of objects [5]. Subsequently, it is shown that from a RST viewpoint, we can associate a natural indiscernibility relation with a formal context. Hence, for the purpose of concept approximation, it is unnecessary to seek an external equivalence relation provided by experts [6]. Unlike approximations based on equivalence relations, our approach is based on the preorder associated with a formal context.

Recognizing that not all equivalence classes of the indiscernibility relation align with extents of formal concepts, a suggestion in [7] proposes the use of meet-irreducible and join-irreducible concepts as fundamental building blocks for rough concept approximations. However, the pivotal theorems (Theorems 4.3 and 4.5) in [7] inaccurately equate join-irreducible and meet-irreducible concepts with object and attribute concepts, respectively. In contrast, our characterization results in Section 3.1 demonstrate the inaccuracy of their claims, offering a proper criterion for identifying irreducible concepts within a concept lattice. Building on this, we advocate the use of extents of object concepts or intents of attribute concepts as the foundational building blocks for rough conceptual approximations.

Furthermore, ref. [8] introduces object-oriented and ref. [9] introduces attribute-oriented concepts. Distinct from the object and attribute concepts considered in this paper, object-oriented and attribute-oriented concepts do not fit the formal concepts paradigm of FCA. Instead, they resemble rough approximations between two universes. For instance, in a formal context $(G,M,I)$, a pair $(A,B)\in \wp (G)\times \wp (M)$ is deemed an object-oriented concept if B is the lower approximation of A, and A is the upper approximation of B. This nuanced distinction adds depth to the conceptual landscape, underscoring the diverse perspectives within the broader realms of FCA and RST.

2. Preliminaries

If U is a finite set, the number of its elements is symbolically denoted by $\left|U\right|$. Let $R\subseteq U\times V$ be a binary relation from a set U to another set V. Then, following Norris [10], for any $u\in U$ and $v\in V$, we use the notations

$$uR=\{v\in V\mid uRv\},Rv=\{u\in U\mid uRv\}.$$

(3)

2.1. Preorders, Partial Orders, and Equivalence Relations

A (binary) relation R on a set U, i.e., $R\subseteq U\times U$, is called a preorder if it is reflexive and transitive. A symmetric (respectively, anti-symmetric) preorder is called an equivalence relation (respectively, a partial order).

We will use the symbol ≤ to denote a partial order on a set U, and call $(U,\le )$ a partially ordered set, or a poset. We write $x<y$ for “$x\le y$ and $x\ne y$”. The relation ≥ on U is defined by $y\ge x$ if and only if (shortened to iff) $x\le y$ is the inverse relation of ≤.

For elements x and y of a given poset $(U,\le )$, we say x is covered by y or y covers x, written $x\prec y$ or $y\succ x$, if $x<y$ and there exists no element $z\in U$ such that $x<z<y$. In this case, x is called a lower cover of y, and y is an upper cover of x.

If R is an equivalence relation on a set U and $u\in U$, then the R-equivalence class of u, denoted ${\left[u\right]}_R$, is the set of all elements $u^{\prime }\in U$ such that $(u,u^{\prime })\in R$. The set of all distinct R-equivalence classes, denoted $U/R$, is called the quotient set of U modulo R.

Lemma 1 

([11]). A preorder ⊑ on a set U determines a topology $\mathcal{T}$ whose basis is the family of all sets $N_u=\{v\mid u⊑v.\}$ for any $u\in U$. In addition, $(U,\mathcal{T})$ is an Alexandroff space.

Lemma 2 

([12]). If f is a function with domain U, then the relation~on U defined by $x\sim y$ iff $f(x)=f(y)$ is an equivalence relation on U, called the kernel relation of f.

2.2. Lattices

A poset $(L,\le )$ is called a lattice if every pair of elements x and y in L has both a meet (greatest lower bound) $x\wedge y$ and a join (least upper bound) $x\vee y$.

If a lattice has a least (respectively, largest) element, then it is unique. It is called the bottom (respectively, top) element of the lattice. We are especially interested in finite lattices. The bottom and top elements exist for every nonempty finite lattice.

Let $(L,\le )$ be a nonempty finite lattice with ⊥ and ⊤ representing its bottom and top elements. An atom of $(L,\le )$ is an element covering ⊥. Dually, a coatom is an element covered by ⊤. An element $j\in L\setminus \{\perp \}$ is join-irreducible if whenever $j=x\vee y$ for $x,y\in L$, then $j=x$ or $j=y$. Dually, an element $m\in L\setminus \{\top \}$ is meet-irreducible if whenever $m=x\wedge y$ for $x,y\in L$, then $m=x$ or $m=y$.

Proposition 1 

([3]). An element of a nonempty finite lattice is join-irreducible iff it has exactly one lower cover and meet-irreducible iff it has exactly one upper cover.

Corollary 1. 

In a nonempty finite concept lattice, every atom (respectively, coatom) is join-irreducible (respectively, meet-irreducible).

2.3. Formal Concept Analysis

In a formal context $(G,M,I)$, ${{(}^{\uparrow })}^{\downarrow }:\wp (G)⟶\wp (G)$ and ${{(}^{\downarrow })}^{\uparrow }:\wp (M)⟶\wp (M)$ are closure operators on G and M, respectively. A closure operator on a set U is a function from the power set of U into itself which is extensive, isotonic (order-preserving), and idempotent. Furthermore, as can be easily seen from [7], for each $g\in G$ and $m\in M$,

$${({\left\{g\right\}}^{\uparrow })}^{\downarrow }={(gI)}^{\downarrow }=\{g^{\prime }\in G\mid g^{\prime }I\supseteq gI\},$$

(4)

$${({\left\{m\right\}}^{\downarrow })}^{\uparrow }={(Im)}^{\uparrow }=\{m^{\prime }\in M\mid I{m}^{\prime }\supseteq Im\}.$$

(5)

$\mathrm{Ext}(G,M,I)$ denotes the set of all extents of concepts in $(G,M,I)$. We then have

$$\mathrm{Ext}(G,M,I)=\{A\subseteq G\mid A={(A^{\uparrow })}^{\downarrow }\}=\{{(X^{\uparrow })}^{\downarrow }\mid X\subseteq G\}$$

(6)

Since ${{(}^{\uparrow })}^{\downarrow }:\wp (G)⟶\wp (G)$ is extensive and isotonic, it follows that for $X\subseteq G$,

$$\begin{array}{ccc}\hfill X\subseteq A\in \mathrm{Ext}(G,M,I)& ⟹& X\subseteq {(X^{\uparrow })}^{\downarrow }\subseteq {(A^{\uparrow })}^{\downarrow }\hfill \\ & ⟹& {(X^{\uparrow })}^{\downarrow }\subseteq A\hfill \end{array}$$

(7)

This gives:

Lemma 3. 

Let $(G,M,I)$ be a formal context. Then

1. 

For each $X\subseteq G$, ${(X^{\uparrow })}^{\downarrow }$ is the smallest extent containing X.

2. 

For each $g\in G$ and $A\in \mathrm{Ext}(G,M,I)$,

$$g\in A⟹{({\left\{g\right\}}^{\uparrow })}^{\downarrow }\subseteq A$$

(8)

Let us recall the basic theorem of FCA.

Theorem 1 

(The basic theorem on concept lattices [3]). Let $(G,M,I)$ be a formal context. Then, $(\mathcal{B}(G,M,I),\le )$ is a complete lattice in which the infimum and supremum of a family of formal concepts $(A_t,B_t)$, $t\in T$, are, respectively, given by

$$\underset{t\in T}{\bigwedge }(A_t,B_t)=(\bigcap _{t\in T}{A}_t,{({(\bigcup _{t\in T}{B}_t)}^{\downarrow })}^{\uparrow }),$$

(9)

$$\underset{t\in T}{\bigvee }(A_t,B_t)=({({(\bigcup _{t\in T}{A}_t)}^{\uparrow })}^{\downarrow },\bigcap _{t\in T}{B}_t).$$

(10)

In addition, for any formal concept $(A,B)\in \mathcal{B}(G,M,I)$, we have

$$\underset{g\in A}{\bigvee }({({\left\{g\right\}}^{\uparrow })}^{\downarrow },{\left\{g\right\}}^{\uparrow })=(A,B)=\underset{m\in B}{\bigwedge }({\left\{m\right\}}^{\downarrow },{({\left\{m\right\}}^{\downarrow })}^{\uparrow }).$$

(11)

In a formal context $(G,M,I)$, the set of its object concepts is $\mathcal{O}(G,M,I)$. Similarly, $\mathcal{A}(G,M,I)$ denotes the set of its attribute concepts. We denote $\mathrm{Int}(G,M,I)$ the set of concept intents of $(G,M,I)$. It is clear from (1) that either the extent or intent of a formal concept can uniquely identify the formal concept. As a consequence, we obtain

$$\mathcal{B}(G,M,I)=\{(B^{\downarrow },B)\mid B\in \mathrm{Int}(G,M,I)\}$$

(12)

Using (3), we have

$${\left\{g\right\}}^{\uparrow }=gI\mathrm{for}\mathrm{each}g\in G$$

(13)

That is, $gI$ is the object intent of g and so, by (11),

$$\mathrm{Int}(G,M,I)=\{\bigcap _{gI\in \mathcal{C}}gI\mid \mathcal{C}\subseteq \{gI\mid g\in G\}\}$$

(14)

From (1) and (13), we have for any $g,g^{\prime }\in G$,

$${({\left\{g^{\prime }\right\}}^{\uparrow })}^{\downarrow }={({\left\{g\right\}}^{\uparrow })}^{\downarrow }⟺g^{\prime }I=gI$$

(15)

Dually, we have

$${\left\{m\right\}}^{\downarrow }=Im\mathrm{for}\mathrm{each}m\in M,$$

(16)

and for any $m,m^{\prime }\in M$,

$${({\left\{m^{\prime }\right\}}^{\downarrow })}^{\uparrow }={({\left\{m\right\}}^{\downarrow })}^{\uparrow }⟺I{m}^{\prime }=Im$$

(17)

$R_M$ denotes the kernel relation of the function $g⟼gI$ on G. That is, $R_M$ is the indiscernibility relation $ind(M)$ if we regard $(G,M,I)$ as an information system. We then have

$${({\left\{g\right\}}^{\uparrow })}^{\downarrow }=\bigcup \{{\left[g^{\prime }\right]}_{R_M}\mid g^{\prime }I\supseteq gI\}\mathrm{for}\mathrm{each}g\in G,$$

(18)

and therefore, we have

$${\left[g\right]}_{R_M}\subseteq {({\left\{g\right\}}^{\uparrow })}^{\downarrow }\mathrm{for}\mathrm{each}g\in G,$$

(19)

This, together with (8), implies that every extent of $(G,M,I)$ is a definable set in the approximation space $(U,R_M)$. In other words,

$$\underline{R_M}(A)=A=\overline{R_M}(A)\mathrm{for}\mathrm{each}A\in \mathrm{Ext}(G,M,I).$$

(20)

Dually, let $R_G$ be the kernel relation of the function $m⟼Im$ on M. Then we have

$${({\left\{m^{\prime }\right\}}^{\downarrow })}^{\uparrow }=\bigcup \{{\left[m^{\prime }\right]}_{R_G}\mid I{m}^{\prime }\supseteq Im\}\mathrm{for}\mathrm{each}m\in M,$$

(21)

$${\left[m\right]}_{R_G}\subseteq {({\left\{m\right\}}^{\downarrow })}^{\uparrow }\mathrm{for}\mathrm{each}m\in M$$

(22)

2.4. Rough Set Theory

The basic construct of RST is the (Pawlakian) approximation space $(U,R)$, where U is a finite set of objects and R is an equivalence relation on U. For any $X\subseteq U$, its lower and upper approximations, $\underline{R}X$ and $\overline{R}X$, in the approximation space are, respectively, defined as

$$\underline{R}X:=\{u\mid {\left[u\right]}_R\subseteq X\},$$

(23)

$$\overline{R}X:=\{u\mid {\left[u\right]}_R\cap X\ne \varnothing \}.$$

(24)

The R-equivalence classes are called elementary sets. A set $X\subseteq U$ is called R-definable if and only if it is a finite union of elementary sets. In this case, $\underline{R}X=\overline{R}X$.

While the Pawlakian approximation space is based on the equivalence relation, there has been extensive work on generalized approximation space defined as $(U,R)$, where R is an arbitrary binary relation on U. For such a generalized approximation space, the lower and upper approximations of a subset $X\subseteq U$ are

$$\underline{R}X:=\{u\mid uR\subseteq X\},$$

(25)

$$\overline{R}X:=\{u\mid uR\cap X\ne \varnothing \}.$$

(26)

As RST provides an effective tool for data analysis; an approximation space is usually induced from a data table formally defined as an information system [2].

Definition 1. 

An information system is a quadruple

$$(U,A,\{V_i\mid i\in A\},\{f_i\mid i\in A\}),$$

where

• U is a finite set, called the universe;
• A is a finite set of attributes;
• for each $i\in A$, $V_i$ is the domain of values for i;
• for each $i\in A$, $f_i:U⟶V_i$ is a total function.

Given an information system $\mathfrak{I}=(U,A,\{V_i\mid i\in A\},\{f_i\mid i\in A\})$ and a subset of attributes $B\subseteq A$, the indiscernibility relation with respect to B, denoted by $ind(B)$, is a binary relation on U defined by

$$(x,y)\in ind(B)⟺f_i(x)=f_i(y)\forall i\in B.$$

Sometimes, we also write $ind(\mathfrak{I})$ to denote the indiscernibility relation $ind(A)$ with respect to all attributes. Obviously, $ind(B)$ is an equivalence relation on U, and so $(U,ind(B))$ is an approximation space.

A straightforward connection between FCA and RST is that a formal context can be regarded as a special kind of information system. Formally, the incidence matrix of the binary relation I in a formal context $(G,M,I)$ is simply a two-valued information system $(U,A,\{V_i\mid i\in A\},\{f_i\mid i\in A\})$ such that $U=G,A=M,V_a=\{0,1\}$ for all $a\in M$, and $f_a(x)=1$ iff $(x,a)\in I$ for all $x\in G$ and $a\in M$. With this viewpoint, the set of objects G and the indiscernibility relation $ind(M)$ form an approximation space. Dually, the transposed incidence matrix of I is also a two-valued information system with rows and columns of the above-mentioned information system being exchanged. Then, $(M,ind(G))$ is also an approximation space. Hence, rough set approximations can be defined for subsets of objects or attributes in a formal context. This is exactly the approach of formal rough concept analysis proposed in [6].

3. Ordered Sets from Formal Contexts

Analogous to the relationship between approximation spaces and information systems, in this section, we show the correspondence between preordered sets and formal contexts. The correspondence can be constructed from the perspective of objects or attributes. Here, we only consider the perspective of objects as all results can be easily dualized to the attribute case.

On the one hand, observing from (4), we can define a preorder ${⊑}_{\mathfrak{C}}$ on G for a formal context $\mathfrak{C}=(G,M,I)$ as follows:

$$g{⊑}_{\mathfrak{C}}{g}^{\prime }:⟺gI\subseteq g^{\prime }I(⟺{({\left\{g^{\prime }\right\}}^{\uparrow })}^{\downarrow }\subseteq {({\left\{g\right\}}^{\uparrow })}^{\downarrow }).$$

(27)

On the other hand, given a preordered set $\mathfrak{P}=(G,⊑)$, we can define a canonical formal context ${\mathfrak{C}}_{\mathfrak{P}}=(G,G,⊒)$. In other words, it is a special kind of formal context $(G,M,I)$, where $M=G$ and $gI{g}^{\prime }$ iff $g^{\prime }⊑g$. Then, it is easy to show that ${⊑}_{{\mathfrak{C}}_{\mathfrak{P}}}=⊑$.

For a preordered set $\mathfrak{P}=(G,⊑)$, the binary relation ≡ defined by $x\equiv y$ iff $x⊑y\wedge y⊑x$ is an equivalence relation. In fact, ≡ is a congruence relation with respect to ⊑. Hence, a preordered set $(G,⊑)$ naturally determine a poset $(G/\equiv ,\le )$, where ≤ is defined by ${\left[x\right]}_{\equiv }\le {\left[y\right]}_{\equiv }$ iff $x⊑y$.

Based on the viewpoint of formal contexts as information systems mentioned in Section 2.4, it is easy to verify that the equivalence relation ${\equiv }_{\mathfrak{C}}$ induced from ${⊑}_{\mathfrak{C}}$ is exactly the indiscernibility relation $ind(\mathfrak{C})$. In addition, the poset determined by ${⊑}_{\mathfrak{C}}$ is isomorphic to $(\{gI\mid g\in G\},\subseteq )$.

Next, we study properties of a formal context based on its induced poset and preorder. For the simplicity of presentation, we will fix a formal context $\mathfrak{C}$ and omit the subscript of orders induced from $\mathfrak{C}$. For example, we will write the preorder ${⊑}_{\mathfrak{C}}$ simply as ⊑.

3.1. Join-Irreducibles and Meet-Irreducibles of Finite Concept Lattices

Recall that two posets $(P,{\le }_P)$ and $(Q,{\le }_Q)$ are isomorphic (respectively, dual isomorphic) if there exists a function f from P onto Q such that $x{\le }_{P}y$ iff $f(x){\le }_{Q}f(y)$ (respectively, $f(y){\le }_{Q}f(x)$). In this case, f is bijective and is called an order isomorphism (respectively, a dual order isomorphism).

In a formal context $(G,M,I)$, it is clear from (1), (2), and Theorem 1 that $\mathrm{Ext}(G,M,I)$ and $\mathrm{Int}(G,M,I)$ are lattices under inclusion. $\mathcal{O}(G,M,I)$ and $\mathcal{A}(G,M,I)$ are posets under the order ≤ from the concept lattice. Therefore, from (2) and (13), we obtain the following

Lemma 4. 

Let $(G,M,I)$ be a formal context. Then

1. 

$(\mathrm{Ext}(G,M,I),\subseteq )$ is a lattice isomorphic to the concept lattice $(\mathcal{B}(G,M,I),\le )$.

2. 

$(\mathrm{Int}(G,M,I),\subseteq )$ is a lattice dual isomorphic to the concept lattice $(\mathcal{B}(G,M,I),\le )$.

3. 

The posets $(\mathcal{O}(G,M,I),\le )$ and $(\{gI\mid g\in G\},\subseteq )$ are dual isomorphic.

4. 

The posets $(\mathcal{A}(G,M,I),\le )$ and $(\{Im\mid m\in M\},\subseteq )$ are isomorphic.

In the remainder of this section, we assume that $(G,M,I)$ is a formal context whose concept lattice $\mathcal{B}(G,M,I)$ is finite. From (10) and the principle of vacuous truth, the extent of the top element ⊤ is (always) the object set G. Dually, the intent of the bottom element ⊥ is the attribute set M. Using (3), we have $G^{\uparrow }={\displaystyle \bigcap _{g\in G}}gI$ and $M^{\downarrow }={\displaystyle \bigcap _{m\in M}}Im$. It follows that

$$\perp =(\bigcap _{m\in M}Im,M),\top =(G,\bigcap _{g\in G}gI)$$

(28)

Notice that the intent $G^{\uparrow }$ of ⊤ can be empty if ${\displaystyle \bigcap _{g\in G}}gI=\varnothing$. Dually, the extent $M^{\downarrow }$ of ⊥ can be empty, if no such object has been specified.

$\mathcal{J}$ denotes the set of join-irreducible elements of $\mathcal{B}(G,M,I)$. From (11), we have

$$\mathcal{J}\subseteq \mathcal{O}(G,M,I)$$

(29)

This, together with (28), Corollary 1, and Theorem 1, leads to the following:

Lemma 5. 

Let $(G,M,I)$ be a formal context whose concept lattice $\mathcal{B}(G,M,I)$ is a finite lattice. An atom of $\mathcal{B}(G,M,I)$ can be characterized in any of the following ways:

1. 

It has the bottom element of $\mathcal{B}(G,M,I)$ as its lower cover.

2. 

It is an object concept whose intent is covered by M in the lattice $(\mathrm{Int}(G,M,I),\subseteq )$.

3. 

It is an object concept whose intent is a maximal element within the poset

$$(\{gI\mid g\in G\}\setminus \{M\},\subseteq ).$$

4. 

It is an object concept whose intent is covered by M within the poset

$$(\{gI\mid g\in G\}\cup \{M\},\subseteq ).$$

This gives characterizations of atoms of finite concept lattices in terms of object intents. Clearly, the dual of Lemma 5 also holds in a finite concept lattice. As a consequence, we obtain the following result.

Lemma 6. 

Let $(G,M,I)$ be a formal context whose concept lattice $\mathcal{B}(G,M,I)$ is a finite lattice. A coatom of $\mathcal{B}(G,M,I)$ can be characterized in any of the following ways:

1. 

It has the top element of $\mathcal{B}(G,M,I)$ as its upper cover.

2. 

It is an attribute concept whose extent is covered by G within the lattice

$$(\mathrm{Ext}(G,M,I),\subseteq ).$$

3. 

It is an attribute concept whose extent is a maximal element within the poset

$$(\{Im\mid m\in M\}\setminus \{G\},\subseteq ).$$

4. 

It is an attribute concept whose extent is covered by G within the poset

$$(\{Im\mid m\in M\}\cup \{G\},\subseteq ).$$

Within the poset $(\{gI\mid g\in G\},\subseteq )$, if an object intent $\tilde{g}I$ of the formal context $(G,M,I)$ has exactly one upper cover, say ${\tilde{g}}^{*}I$, then for any nonempty $\mathcal{F}\subseteq \{gI\mid g\in G\}$,

$$\begin{array}{ccc}\hfill \tilde{g}I\subset \bigcap _{gI\in \mathcal{F}}gI& ⟹& \tilde{g}I\subset {\tilde{g}}^{*}I\subseteq gI\mathrm{for}\mathrm{every}gI\in \mathcal{F}\hfill \\ & ⟹& \tilde{g}I\subset {\tilde{g}}^{*}I\subseteq \bigcap _{gI\in \mathcal{F}}gI\hfill \end{array}$$

(30)

This, together with Lemma 5, implies that an object intent has exactly one upper cover within the lattice $(\mathrm{Int}(G,M,I),\subseteq )$ iff it has exactly one upper cover within the poset $(\{gI\mid g\in G\}\cup \{M\},\subseteq )$. Since the lattice $(\mathrm{Int}(G,M,I),\subseteq )$ and the concept lattice are dual isomorphic, and since the posets $(\mathcal{O}(G,M,I),\le )$ and $(\{gI\mid g\in G\},\subseteq )$ are dual isomorphic, we obtain the following:

Proposition 2. 

Let $(G,M,I)$ be a formal context whose concept lattice $\mathcal{B}(G,M,I)$ is a finite lattice. A formal concept is join-irreducible iff it is an object concept whose intent having exactly one upper cover within the poset $(\{gI\mid g\in G\}\cup \{M\},\subseteq )$.

Dually, we have the following:

Proposition 3. 

Let $(G,M,I)$ be a formal context whose concept lattice $\mathcal{B}(G,M,I)$ is a finite lattice. A formal concept is meet-irreducible iff it is an attribute concept whose attribute extent has exactly one upper cover within the poset $(\{Im\mid m\in M\}\cup \{G\},\subseteq )$.

Propositions 2 and 3 furnish criteria for establishing the join-irreducibility of an object concept and the meet-irreducibility of an attribute concept, respectively. These propositions reveal a crucial insight: not all object concepts are join-irreducible, and, similarly, not all attribute concepts are meet-irreducible. This observation underscores a limitation in coverage when relying solely on irreducible concepts as the foundational elements for rough approximations. Recognizing this coverage gap, we present a solution in the subsequent subsection. To overcome the limitations posed by the exclusive use of irreducible concepts, we put forth a novel definition of rough conceptual approximation, offering a more inclusive and robust approach to address the issue of coverage comprehensively.

3.2. Rough Conceptual Approximations

According to Lemma 1, the relation ⊑ gives rise to an Alexandroff topology $\mathcal{T}$ on the set G. Within this topological framework, for each element $g\in G$, the set $N_g=\{g^{\prime }\mid g⊑g^{\prime }\}=\{g^{\prime }\mid gI\subseteq g^{\prime }I\}={({\left\{g\right\}}^{\uparrow })}^{\downarrow }$ emerges as the smallest open neighborhood of g. This formulation is intuitively grasped, as it essentially captures the essence of the binary neighborhood system [13], denoted $BN:U⟶\wp (U)$, defined on U, where $BN(g)={({\left\{g\right\}}^{\uparrow })}^{\downarrow }$. Notably, this binary neighborhood system is equivalent to the topological neighborhood system of the corresponding Alexandroff space $(G,\mathcal{T})$. Motivated by the concepts of topological closure and interior operators within the Alexandroff space $(G,\mathcal{T})$, we introduce the notion of rough conceptual approximations. This innovative conceptualization draws inspiration from the rich foundation of topological considerations, enhancing our understanding and operationalization of rough approximations within the context of formal concepts.

Definition 2. 

Let $(G,M,I)$ be a formal context. For any $X\subseteq G$, the lower conceptual approximation of X, denoted by ${I_M}_{*}(X)$, and upper conceptual approximation of X, denoted by ${I_M}^{*}(X)$, are, respectively, defined as follows:

$${I_M}_{*}(X)=\{g\in G\mid {({\left\{g\right\}}^{\uparrow })}^{\downarrow }\subseteq X\},$$

(31)

$${I_M}^{*}(X)=\{g\in G\mid {({\left\{g\right\}}^{\uparrow })}^{\downarrow }\cap X\ne \varnothing \}.$$

(32)

Using (18), (31) and (32) can be rewritten as

$${I_M}_{*}(X)=\bigcup _{g\in G:{({\left\{g\right\}}^{\uparrow })}^{\downarrow }\subseteq X}{\left[g\right]}_{R_M}$$

(33)

$${I_M}^{*}(X)=\bigcup _{g\in G:{({\left\{g\right\}}^{\uparrow })}^{\downarrow }\cap X\ne \varnothing }{\left[g\right]}_{R_M}.$$

(34)

From the definition, we can derive the following theorem.

Theorem 2. 

If $(G,M,I)$ is a formal context $(G,M,I)$, then

1. 

The lower and upper conceptual approximations

$${I_M}_{*}:G⟶\wp (G)\mathrm{and}{I_M}^{*}:G⟶\wp (G)$$

are the topological interior and closure operators, respectively, on $G;$

2. 

The set $\mathcal{T}=\{A\subseteq G\mid {I_M}_{*}(A)=A\}$ is an Alexandroff topology on $G;$

3. 

For each $g\in G$, ${({\left\{g\right\}}^{\uparrow })}^{\downarrow }$ is the smallest open neighborhood of g in the Alexandroff space $(G,\mathcal{T})$.

According to Theorem 2, we have ${I_M}_{*}(X)\subseteq X$ for any $X\subseteq G$. This, together with (8) and (31), leads to the following:

Corollary 2. 

If $(G,M,I)$ is a formal context $(G,M,I)$, then

$${I_M}_{*}(A)=A\mathrm{for}\mathrm{each}A\in \mathrm{Ext}(G,M,I)$$

(35)

Combining (20) and (35), we obtain

$${I_M}_{*}(A)=\underline{R_M}(A)=A=\overline{R_M}(A)\mathrm{for}\mathrm{each}A\in \mathrm{Ext}(G,M,I)$$

(36)

Consider $R_M$ and ⊑ as the indiscernibility relation and preorder, respectively, associated with the formal context $(G,M,I)$. Notably, $(G,R_M)$ and $(G,⊑)$ form a Pawlakian and generalized approximation space, respectively. Consequently, it is a natural progression to define lower and upper approximations within these two spaces leveraging RST. The concept of rough set approximations for FCA has been established using the approximation space $(G,R_M)$ in prior work [6]. In this context, our definition of rough conceptual approximations, inspired by topology and encapsulated in the definition of $N_g$, seamlessly aligns with the standard definition of rough set approximations within the space $(G,⊑)$. To formalize this alignment, we present the following proposition, establishing the equivalence between our topology-inspired definition and the conventional definition of rough set approximations in the space $(G,⊑)$.

Proposition 4. 

Let $(G,⊑)$ be a generalized approximation space in which lower and upper approximations are defined by

$$\underline{⊑}(X)=\{x\mid \forall y(x⊑y\to y\in X\},$$

$$\overline{⊑}(X)=\{x\mid \exists y(x⊑y\wedge y\in X\}.$$

Then, ${I_M}_{*}(X)=\underline{⊑}(X)$ and ${I_M}^{*}(X)=\overline{⊑}(X)$.

As $R_M$ is a sub-relation of ⊑, the following relationship follows immediately.

Proposition 5. 

For a formal context $(G,M,I)$ and any $X\subseteq G$,

$${I_M}_{*}(X)\subseteq \underline{R_M}(X)\subseteq X\subseteq \overline{R_M}(X)\subseteq {I_M}^{*}(X)$$

(37)

4. Discussion

4.1. Illustrative Examples

In this section, we provide two illustrative examples within a formal context. These examples serve to both demonstrate and validate the results obtained.

Example 1. 

Consider a formal context denoted as $(G,M,I)$, where $G=\{g_1,\dots ,g_7\}$, and $M=\{m_1,\dots ,m_4\}$. The incidence relation I is elucidated in

Table 1. Exemplary formal context.

I	${\mathit{m}}_1$	${\mathit{m}}_2$	${\mathit{m}}_3$	${\mathit{m}}_4$
$g_1$	0	1	1	0
$g_2$	1	0	1	1
$g_3$	0	1	1	1
$g_4$	0	1	1	1
$g_5$	1	0	1	1
$g_6$	1	1	0	1
$g_7$	0	1	0	0

: a “1” at the intersection of row $g_i$ and column $m_j$ signifies that the object $g_i$ possesses attribute $m_j$, while a “0” implies that $(g_i,m_j)\notin I$.

In reference to Table 1, it is evident that

$$\begin{array}{ccc}\hfill & & g_{1}I=\{m_2,m_3\},g_{2}I=g_{5}I=\{m_1,m_3,m_4\},\hfill \\ & & g_{3}I=g_{4}I=\{m_2,m_3,m_4\},\hfill \\ & & g_{6}I=\{m_1,m_2,m_4\},g_{7}I=\left\{m_2\right\}.\hfill \end{array}$$

(38)

Utilizing $(10)$, $(11)$ and $(38)$, we can compute the concept intents of the formal context $(G,M,I)$ as shown in Figure 1.

Furthermore, applying $(12)$ in conjunction with Figure 1, we are able to derive the formal concepts of $(G,M,I)$ as illustrated in Figure 2.

Examining Figure 2, it becomes evident that the object concepts $C_7=(\{g_3,g_4\},\{m_2,m_3,m_4\})$, $C_8=(\left\{g_6\right\},\{m_1,m_2,m_4\})$, and $C_9=(\{g_2,g_5\},\{m_1,m_3,m_4\})$ are the atoms of the concept lattice. By referring to $(38)$ and consulting Figure 1, we ascertain that the intents $\{m_2,m_3,m_4\}$, $\{m_1,m_2,m_4\}$, and $\{m_1,m_3,m_4\}$ corresponding to $C_7$, $C_8$, and $C_9$, respectively, are

1. 

The elements covered by M in the lattice $(\mathrm{Int}(G,M,I),\subseteq )$.

2. 

The maximal elements of the poset $(\{g_{j}I\mid g_j\in G\}\setminus \left\{M\right\},\subseteq )$.

3. 

The elements covered by M within the poset $(\{g_{j}I\mid g_j\in G\}\cup \left\{M\right\},\subseteq )$.

These observations serve to confirm the validity of Lemma 5.

Within the concept lattice, Figure 2 yields the following noteworthy observations:

1. 

$C_4$ and $C_5$ are the lower covers of $C_1$.

2. 

$C_4$ and $C_6$ are the lower covers of $C_2$.

3. 

$C_8$ and $C_9$ are the lower covers of $C_3$.

4. 

$C_4=(\{g_1,g_3,g_4\},\{m_2,m_3\})$ has $C_7=(\{g_3,g_4\},\{m_2,m_3,m_4\})$ as its only lower cover.

5. 

$C_7$ and $C_8$ are the lower covers of $C_5$.

6. 

$C_7$ and $C_9$ are the lower covers of $C_6$.

It is evident that, with the exception of the atoms, the object concept $C_4$ stands alone as the sole element possessing precisely one lower cover within the concept lattice. Further, through reference to $(38)$, it becomes apparent that, aside from the intents of the atoms, the object intent $\{m_2,m_3\}$ of $C_4$ represents the only object intent characterized by exactly one upper cover within the poset $(\{g_{j}I\mid g_j\in G\}\cup \left\{M\right\},\subseteq )$.

In this illustrative example, we commence our analysis with the foundational insights provided by Lemma 5. This lemma offers profound characterizations of atoms within finite concept lattices, particularly in relation to object intents. Building upon this groundwork, we proceed to delve into Proposition 2, a pivotal proposition that articulates a critical criterion for join-irreducibility within a formal context $(G,M,I)$ possessing a finite concept lattice. According to Proposition 2, a formal concept achieves join-irreducibility precisely when it is an object concept, and its object intent stands as the sole element positioned above it within the poset $(\{gI\mid g\in G\}\cup \{M\},\subseteq )$. This proposition establishes a clear and concise condition for join-irreducibility, contributing to the nuanced understanding of the interplay between object concepts and their corresponding object intents within the broader context of formal concept lattices.

Example 2. 

Let us proceed with Example 1. Referring to equation $(38)$, we find that

$$U/R_M=\{\left\{g_1\right\},\{g_2,g_5\},\{g_3,g_4\},\left\{g_6\right\},\left\{g_7\right\}\}.$$

(39)

Utilizing $(4)$, $(38)$, and $(39)$, we are able to calculate ${({\left\{g\right\}}^{\uparrow })}^{\downarrow }$ and ${\left[g\right]}_{R_M}$ for each object $g\in G$ as outlined in

Table 2. ${({\left\{g\right\}}^{\uparrow })}^{\downarrow }$ and ${\left[g\right]}_{R_M}$ for every $g\in G$.

$\mathbf{Object}\mathit{g}$	${({\left\{\mathit{g}\right\}}^{\uparrow })}^{\downarrow }$	${\left[\mathit{g}\right]}_{{\mathit{R}}_{\mathit{M}}}$
$g_1$	$\{g_1,g_3,g_4\}$	$\left\{g_1\right\}$
$g_2$	$\{g_2,g_5\}$	$\{g_2,g_5\}$
$g_3$	$\{g_3,g_4\}$	$\{g_3,g_4\}$
$g_4$	$\{g_3,g_4\}$	$\{g_3,g_4\}$
$g_5$	$\{g_2,g_5\}$	$\{g_2,g_5\}$
$g_6$	$\left\{g_6\right\}$	$\left\{g_6\right\}$
$g_7$	$\{g_1,g_3,g_4,g_6,g_7\}$	$\left\{g_7\right\}$

.

From Table 2, it is evident that ${\left[g_j\right]}_{R_M}\subseteq {({\left\{g_j\right\}}^{\uparrow })}^{\downarrow }$ for every $g_j\in G$. These observations serve to validate the assertion presented in $(19)$.

Let us estimate the rough set and rough conceptual approximations for the extent $A=\{g_1,g_3,g_4\}$, the set $X=\{g_1,g_2,g_3,g_5\}\notin \mathrm{Ext}(G,M,I)$, and the set $Y=\{g_1,g_2,g_3,g_5,g_6\}\notin \mathrm{Ext}(G,M,I)$. Referring to $(23)$, $(24)$, and Table 2, we obtain

$$\underline{R_M}(A)=\{g_1,g_3,g_4\},\overline{R_M}(A)=\{g_1,g_3,g_4\}$$

(40)

$$\underline{R_M}(X)=\{g_1,g_2,g_5\},\overline{R_M}(X)=\{g_1,g_2,g_3,g_4,g_5\}$$

(41)

$$\underline{R_M}(Y)=\{g_1,g_2,g_5,g_6\},\overline{R_M}(Y)=\{g_1,g_2,g_3,g_4,g_5,g_6\}$$

(42)

In accordance with $(31)$, $(32)$, and the information provided in Table 2, we obtain

$${I_M}_{*}(A)=\{g_1,g_3,g_4\},{I_M}^{*}(A)=\{g_1,g_3,g_4,g_7\}$$

(43)

$${I_M}_{*}(X)=\{g_2,g_5\},{I_M}^{*}(X)=\{g_1,g_2,g_3,g_4,g_5,g_7\}$$

(44)

$${I_M}_{*}(Y)=\{g_2,g_5,g_6\}\},{I_M}^{*}(Y)=\{g_1,g_2,g_3,g_4,g_5,g_6,g_7\}$$

(45)

By merging $(40)$ and $(43)$, we derive

${I_M}_{*}(A)=\underline{R_M}(A)=A=\overline{R_M}(A)$.

Combining $(41)$ and $(44)$, we obtain

$$\begin{array}{ccc}& & {I_M}_{*}(X)=\{g_2,g_5\}\subseteq \underline{R_M}(X)=\{g_1,g_2,g_5\}\subseteq X=\{g_1,g_2,g_3,g_5\}\hfill \\ & \subseteq & \overline{R_M}(X)=\{g_1,g_2,g_3,g_4,g_5\}\subseteq {I_M}^{*}(X)=\{g_1,g_2,g_3,g_4,g_5,g_7\}.\hfill \end{array}$$

Combining $(42)$ and $(45)$, we obtain

$$\begin{array}{ccc}\hfill {I_M}_{*}(Y)=\{g_2,g_5,g_6\}& \subseteq & \underline{R_M}(Y)=\{g_1,g_2,g_5,g_6\}\hfill \\ & \subseteq & Y=\{g_1,g_2,g_3,g_5,g_6\}\hfill \\ & \subseteq & \overline{R_M}(Y)=\{g_1,g_2,g_3,g_4,g_5,g_6\}\hfill \\ & \subseteq & {I_M}^{*}(Y)=\{g_1,g_2,g_3,g_4,g_5,g_6,g_7\}.\hfill \end{array}$$

This serves to confirm the validity of (36) and Proposition 5.

Examining Table 1 and Table 2 more closely, we can observe that:

$$\begin{array}{ccc}\hfill ({({\left\{g_7\right\}}^{\uparrow })}^{\downarrow },g_{7}I)& =& (\{g_1,g_3,g_4,g_6,g_7\},\left\{m_2\right\})\hfill \\ & =& (I{m}_2,{({\left\{m_2\right\}}^{\downarrow })}^{\uparrow }).\hfill \end{array}$$

(46)

This revelation highlights the versatile role of the formal concept $(\{g_1,g_3,g_4,g_6,g_7\},$$\left\{m_2\right\})$. It not only functions as the object concept associated with $g_7$ but also serves as the attribute concept linked with symptom $m_2$. Moreover, a meticulous verification confirms that $(\{g_1,g_3,g_4,g_6,g_7\},\left\{m_2\right\})$ holds the distinction of being a coatom within the concept lattice. Consequently, it logically follows that ${({\left\{g_7\right\}}^{\uparrow })}^{\downarrow }=\{g_1,g_3,g_4,g_6,g_7\}$ stands out as a maximal element within $\mathrm{Ext}(G,M,I)\setminus \left\{G\right\}$. Transitioning to Table 1, representing a formal context denoted by $(G,M,I)$ with seven patients $g_1,g_2,g_3,g_4,g_5,g_6$, and $g_7$, and an attribute set M featuring Fever $m_1$, Coughing $m_2$, Aches $m_3$, and Runny nose $m_4$, our formulation succinctly encapsulates the diverse symptoms displayed by these patients. In alignment with (31) and (45), we derive significant insights:

(1)

“$g_2\in {I_M}_{*}(Y)$” (or “$g_5\in {I_M}_{*}(Y)$”) implies “Patients exhibit symptoms including a range of discomforts: {Fever, Aches, Running nose}”.

(2)

“$g_6\in {I_M}_{*}(Y)$” implies ”Patients exhibit symptoms including a range of discomforts: {Fever, Coughing, Running nose}”.

(3)

The extent of formal concept $(\{g_2,g_5,g_6\},\{m_1,m_4\})$ is represented by the lower conceptual approximation ${I_M}_{*}(Y)=\{g_2,g_5,g_6\}$. The intent $\{m_1,m_4\}$ corresponds to the intersection of two sets of symptoms, namely,

$$\left\{\mathrm{Fever},\mathrm{Aches},\mathrm{Runny}\mathrm{nose}\right\}\mathrm{and}\left\{\mathrm{Fever},\mathrm{Coughing},\mathrm{Runny}\mathrm{nose}\right\}.$$

This insight enriches our understanding of the symptomatic profiles, emphasizing the shared symptoms among specific patient groups.

In this comprehensive analysis, we have delved into a set G comprising seven objects labeled as $g_1,g_2,g_3,g_4,g_5,g_6$, and $g_7$. Each of these objects is intricately associated with its respective object intents, representing valuable patient data in the context of specific symptoms. The primary objective of these object intents is to discern the set of patients manifesting particular symptoms, such as Fever, Coughing, Runny nose ($g_{6}I$), captured by ${({\left\{g_6\right\}}^{\uparrow })}^{\downarrow }$. Our focus is on understanding the intersections of these sets with Y.

For example, considering $g_1$, it falls within the upper approximation of Y. This is because the set of patients with symptoms precisely matching $g_{1}I$, namely $\left\{Coughing,Aches\right\}$, is $\left\{g_1\right\}$, making it a part of Y. However, it is crucial to note that the set of patients with symptoms including Coughing and Aches is $\{g_1,g_3,g_4\}$, and this set is not part of Y. On the other hand, $g_7$ is part of the upper conceptual approximation of Y. The set of patients exhibiting symptoms, including Coughing ($g_{7}I$), is $\{g_1,g_3,g_4,g_6,g_7\}$, which has a nonempty intersection with Y. However, the set of patients with symptoms precisely matching $g_{7}I$ is $\left\{g_7\right\}$, and this set is disjointed from Y. This nuanced examination provides a detailed understanding of how specific symptoms, as expressed through object intents, relate to the broader context of patient data, Y, and their intersections, contributing to a richer interpretation of the formal concept lattice.

Overall, this analysis unveils intricate connections between specific symptoms, sets of patients, and the conceptual approximations of Y. Discerning how these sets intersect and interact significantly enhances our comprehension of patient profiles and the underlying patterns of symptoms within the specified context.

This methodical examination of patients’ symptoms not only provides crucial insights for categorizing them based on specific symptom criteria but also lays the groundwork for pioneering innovative clinical diagnostic methods. The systematic approach employed in this analysis can be further expanded to cater to a broader spectrum of patients or adapted for diverse applications beyond the realm of healthcare. Thus, the implications extend beyond the immediate context, opening avenues for impactful advancements in diagnostics and potentially influencing various fields.

4.2. A Comparative Analysis

In Section 1.2, we explored various approaches to the integration of FCA and RST, revealing four distinct definitions of rough approximations based on formal contexts (refs. [5,6,7] and our proposed definition). The crux of the disparities among these definitions lies in the fundamental building blocks employed for approximations. This detailed analysis aims to underscore the implications of such differences, contextualizing our work and elucidating its significant contribution. The definition presented in [5] notably relies on an externally provided equivalence relation, introducing a somewhat loosely coupled integration between RST and FCA. In this framework, given a formal context $(G,M,I)$ and an equivalence relation E over G, the structure can be bifurcated into the formal context itself and an approximation space $(G,E)$, sharing a common universe. For a formal concept $(A,B)$, rough approximations of A are solely defined within the approximation space, devoid of direct dependence on the formal context. However, an equivalence class of E might not align with the extent of a formal concept, limiting its interpretability in natural language. Addressing the issue of loose coupling, the proposal in [6] utilizes an equivalence relation derived from the formal context, fostering a closer connection between RST and FCA. Consequently, rough approximations, i.e., $\underline{R_M}A$ and $\overline{R_M}A$, become more intertwined with the formal context. Yet, despite this improvement, basic building blocks, i.e., equivalence classes of RM, still do not consistently correspond to extents of formal concepts, as highlighted in [7].

To rectify the challenge of considering subsets that lack representation as extents of formal concepts, [7] advocates using extents of irreducible concepts as the foundational elements for rough approximations. Formally defining lower and upper approximations of a subset $A\subseteq G$ as

$$A_{*}=\bigcup \{X\subseteq G\mid (X,X^{\uparrow })\in \mathcal{J}\}$$

and

$$A^{*}=\bigcap \{X\subseteq G\mid (X,X^{\uparrow })\in \mathcal{M}\},$$

respectively, where $\mathcal{J}$ denotes the set of join-irreducible concepts and $\mathcal{M}$ signifies the set of meet-irreducible concepts, while extents of irreducible concepts align with the interpretability framework of FCA, they fall short of providing complete coverage of the concept lattice. Notably, our characterization result reveals that not all object concepts are irreducible. Consequently, a noteworthy implication of the definition in [7] is that $G_{*}$ is not necessarily equal to G, implying the existence of gaps in the set of fundamental building blocks for rough approximations.

Example 3. 

Consider the formal context from Example 1. Its join-irreducible concepts, namely $C_4,C_7,C_8$, and $C_9$, collectively span the extents $\{g_1,g_2,\cdots ,g_6\}$, forming $G_{*}$ as per the aforementioned definition. Notably, $G_{*}$ proves to be a proper subset of G in this specific instance.

In stark contrast, our rough conceptual approximations stand out for their use of extents encompassing all object concepts. This approach not only ensures interpretability but also guarantees comprehensive coverage. Following our definition, we confidently assert $I_{M_{*}}(G)=G=I_{M^{*}}(G)$, spotlighting the exhaustive coverage provided by our fundamental building blocks. Furthermore, it aligns with the principle that all formal concepts in FCA, including object concepts, are inherently interpretable due to their association with names in natural language.

Beyond establishing the theoretical superiority over existing frameworks, our approach holds profound implications for practical applications. Traditional RST-based analyses leverage the interpretability of equivalence classes [14], where each class is defined by its attribute values, expressed through a decision logic (DL) formula ${\bigwedge }_{a\in C}(a,v_a)$, with C denoting the attribute set, a representing an attribute, and $a\in C$ being the attribute value. This approach yields decision rules such as

$$\underset{a\in C}{\bigwedge }(a,v_a)⟶(d,v_X),$$

where ${\bigwedge }_{a\in C}(a,v)$ describes ${\left[u\right]}_R$ and $(d,v_X)$ is an atomic DL formula signifying the decision class X.

Similarly, in FCA, we posit that each formal concept can be articulated through natural language terms, like “Musician”, “Electronic Device”, or “Animal”. Consequently, if $g\in I_{M_{*}}(X)$ for some decision class X, the relation ${({\left\{g\right\}}^{\uparrow })}^{\downarrow }\subseteq X$ corresponds to the decision rule

$${\tau }_g\mathrm{ISA}{\tau }_X,$$

where ${\tau }_g$ and ${\tau }_X$ are term descriptions of the concept ${({\left\{g\right\}}^{\uparrow })}^{\downarrow },{\left\{g\right\}}^{\uparrow })$ and the decision class X, respectively.

The existing definitions of rough approximations pose a challenge in deriving certain decision rules. In formulations relying on equivalence-based definitions, such as those in [5,6], the lack of a guarantee that an equivalence class corresponds precisely to the extent of a formal concept introduces a significant limitation. In essence, the term description by FCA may not be derivable in such cases. Specifically, if a decision class involves an equivalence class $\left[u\right]$ without an associated term description, it becomes impossible to derive an interpretable rule with the antecedent based on the description of $\left[u\right]$. In the context of [5], where an independent approximation space exists, there is a potential for a DL formula describing the equivalence class. Consequently, it becomes feasible to derive an RST-based rule, albeit one that is interpretable yet detached from the formal context. This raises a critical point: when the derivation of rules does not involve the concept lattice, the integration of RST and FCA may lack meaningful significance in such scenarios. Conversely, the definition in [7] stands out as it builds on the fundamental blocks of extents of irreducible concepts. This framework demonstrates the capability to generate rules in a similar form to those derived in our approach. However, its drawback lies in occasional insufficient coverage. For instance, considering the formal context in Example Context, if the decision class encompasses the extent of the concept $C_1$, the definition in [7] fails to derive a rule covering it comprehensively. In contrast, our framework emerges as distinctive in its ability to derive rules with antecedents interpretable through FCA principles while ensuring complete coverage of all object concepts. This unique feature positions our approach as the sole framework capable of providing comprehensive and interpretable rules, addressing the limitations inherent in existing formulations.

In summary, the comparative analysis unequivocally demonstrates the theoretical superiority and practical advantages of our framework over existing approaches. This not only substantiates the significance and value of our contribution but also sets the stage for exploring potential applications stemming from our innovative approach.

4.3. Potential Applications

In the realm of formal contexts, we encounter two primary approaches to rough set approximations: those grounded in equivalence classes denoted as ${\left[g\right]}_{R_M}$ and extents of object concepts represented as ${({\left\{g\right\}}^{\uparrow })}^{\downarrow }$. To elaborate, ${\left[g\right]}_{R_M}$ encompasses all objects sharing precisely the same attributes as g, while ${({\left\{g\right\}}^{\uparrow })}^{\downarrow }$ encompasses objects possessing all attributes of g, potentially with additional attributes beyond those of g.

In practical scenarios, formal contexts serve as a robust framework for dissecting customer behavior within focused marketing endeavors. By utilizing the connections between objects and attributes within a formal context, businesses can discern unique customer segments defined by their common attributes. These segments, whether ${\left[g\right]}_{R_M}$ or ${({\left\{g\right\}}^{\uparrow })}^{\downarrow }$, offer crucial insights into customer behavior and preferences, empowering businesses to formulate more impactful marketing strategies. Here are some potential applications derived from the identified customer segments:

• Personalized Product Recommendations: AI algorithms exhibit a sophisticated capacity to deploy precise techniques such as collaborative filtering or content-based recommendation. This precision enables the identification of products or services most likely to resonate with a specific customer segment. These tailored recommendations, rooted in shared attributes, act as a catalyst for enhancing the overall customer experience. The culmination of this personalized approach invariably leads to a significant boost in conversion rates for businesses [15].
• Churn Prediction and Retention: Harnessing the formidable capabilities of machine learning algorithms opens a gateway to a profound exploration of a business’s historical data. Within this rich tapestry lie distinctive trends and patterns, revealing illuminating attributes that act as beacons in identifying customer segments teetering on the brink of churn. Churn, symbolizing the detachment of customers from the business, becomes a focal point that can be proactively addressed through astute measurement of these pivotal attributes. This proactive approach empowers companies to initiate meticulously tailored retention strategies. These strategies, ranging from enticing discounts and captivating loyalty rewards to intimately personalized incentives, are unified by the shared objective of staunchly preventing churn. By adopting this strategic and data-driven methodology, businesses not only gain insights into potential churn scenarios but also equip themselves to deploy effective and preemptive measures to enhance customer loyalty and overall satisfaction [16,17].
• Cross-Selling and Upselling Opportunities: AI-driven recommender systems possess the capability to intricately analyze a customer segment’s purchase history and behavioral patterns. In doing so, they unveil latent cross-selling and upselling opportunities. By skillfully presenting supplementary products or enticing premium upgrades, strategically grounded in their collective attributes, businesses can substantially elevate their average order value and overall revenue. This strategic approach extends beyond mere financial gains; it concurrently enhances customer satisfaction and fortifies the financial health of the business. Through the intelligent utilization of AI algorithms, companies can not only meet but exceed customer expectations, fostering a more robust and mutually beneficial relationship with their clientele. This fusion of data-driven insights and strategic presentation not only boosts immediate revenue streams but also lays a foundation for sustained growth and customer loyalty in the long term [18].
• Customer Segmentation: Harnessing state-of-the-art clustering algorithms such as K-means or hierarchical clustering, businesses wield the capability to intricately categorize their customer base into well-defined segments, each rooted in shared attributes. The initial cluster assignments, derived from the inherent structure of the data, lay the foundation for artificial intelligence algorithms to further refine these segments. In this iterative process, advanced AI algorithms uncover intricate patterns and subgroups concealed within each segment, revealing nuanced insights. These extracted insights serve as invaluable resources, empowering businesses to craft laser-focused marketing campaigns meticulously tailored to cater to the individualized needs and preferences of each customer segment. Essentially, this approach establishes a dynamic and data-driven framework for personalizing marketing strategies, thereby enhancing customer engagement. The end result is an improvement in customer satisfaction and overall business performance, as businesses adapt and respond more effectively to the diverse and evolving preferences within their customer base [19].

By strategically applying AI techniques, businesses can navigate the intricacies of formal concept analysis, particularly in exploring the extended formal context and the identified equivalence classes or extents. This exploration yields invaluable insights into customer behavior, fostering enhanced customer satisfaction, finely tuned marketing strategies, and ultimately, substantial revenue growth. The genuine power of AI becomes evident as it unravels intricate patterns and discerns emerging trends within extensive datasets. This capability empowers businesses to make informed, data-driven decisions. These decisions, in turn, culminate in the delivery of highly personalized experiences that resonate with customers on a profound level. The fusion of formal concept analysis and AI techniques thus becomes a catalyst for elevating not just customer satisfaction but the overall strategic prowess of businesses in navigating the complex landscape of customer behavior and preferences.

In the realm of potential applications, the efficacy of AI techniques crucially depends on the thoughtful selection of customer segmentation. As previously discussed, there exist at least two viable approaches for assembling a group centered around a particular customer. Specifically, leveraging the equivalence class ${\left[g\right]}_{R_M}$ as a customer segment empowers businesses to precisely target their desired audience with heightened accuracy and reduced costs. Conversely, embracing ${({\left\{g\right\}}^{\uparrow })}^{\downarrow }$ as the customer segment broadens the outreach to encompass a diverse spectrum of potential customers. Hence, the strategic decision between these two approaches should be contingent upon the specific objectives that the business aims to achieve. This meticulous consideration ensures that resources are allocated optimally, thereby maximizing the impact of marketing endeavors and fostering overall success. By aligning the choice of customer segmentation with overarching goals, businesses can enhance the efficiency and effectiveness of their AI-driven marketing strategies, ultimately propelling them towards sustained success in the competitive landscape.

5. Conclusions

FCA and RST stand out as robust mathematical frameworks extensively employed in data analysis and knowledge processing. FCA initiates its exploration by defining a formal context using a triple $(G,M,I)$, where a binary relation establishes connections between objects and attributes. Within this framework, formal concepts emerge as sets encapsulating maximal commonalities between objects and attributes, collectively forming a comprehensive lattice known as the concept lattice. In contrast, RST adopts equivalence relations to partition a set, strategically diminishing the number of attributes under consideration. These partitions lay the groundwork for the approximation space, a pivotal element in ascertaining both lower and upper approximations for a given set. The strategic application of FCA and RST has yielded groundbreaking advancements in knowledge extraction from data, manifesting across diverse domains.

In this paper, we present a set of criteria for identifying atoms and coatoms within finite concept lattices. Additionally, we establish conditions that shed light on specific formal concepts exhibiting join-irreducible or meet-irreducible characteristics. These fundamental properties offer profound insights into the construction and analysis of the lattice encompassing all concepts within a formal context, extending their significance beyond the confines of FCA into diverse domains. More notably, these findings inspire the introduction of a novel concept of rough conceptual approximations, unveiling its equivalence to rough approximations based on the preorder relation. We demonstrate the theoretical superiority of our proposal over existing methods of rough approximations in FCA and underscore its practical applicability to the interpretability of rules induced from RST-based data analysis.

In essence, our contributions mark a substantial leap forward in the fusion of FCA and RST. They arm us with the tools to delve deeper into the intricate structures and relationships within complex datasets, thereby facilitating the extraction of meaningful knowledge with heightened ease and precision. The theoretical advancements presented herein not only expand the boundaries of these mathematical theories but also pave the way for practical applications, promising a more nuanced understanding of complex data landscapes.