# An Order-Theoretic Study on Formal Concept Analysis

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## Abstract

**:**

## 1. Introduction

#### 1.1. Main Contributions

- 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.

#### 1.2. Related Work

## 2. Preliminaries

#### 2.1. Preorders, Partial Orders, and Equivalence Relations

**Lemma 1**

**Lemma 2**

#### 2.2. Lattices

**Proposition 1**

**Corollary 1.**

#### 2.3. Formal Concept Analysis

**Lemma 3.**

- 1.
- For each $X\subseteq G$, ${({X}^{\uparrow})}^{\downarrow}$ is the smallest extent containing X.
- 2.
- For each $g\in G$ and $A\in \mathrm{Ext}(G,M,I)$,$$g\in A\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\u27f9\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{({\left\{g\right\}}^{\uparrow})}^{\downarrow}\subseteq A$$

**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

#### 2.4. Rough Set Theory

**Definition 1.**

- 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 $i\in A$, ${f}_{i}:U\u27f6{V}_{i}$ is a total function.

## 3. Ordered Sets from Formal Contexts

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

**Lemma 4.**

- 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.

**Lemma 5.**

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

**Lemma 6.**

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

**Proposition 2.**

**Proposition 3.**

#### 3.2. Rough Conceptual Approximations

**Definition 2.**

**Theorem 2.**

- 1.
- The lower and upper conceptual approximations$${{I}_{M}}_{*}:G\u27f6\wp (G)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{{I}_{M}}^{*}:G\u27f6\wp (G)$$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 $(G,\mathcal{T})$.

**Corollary 2.**

**Proposition 4.**

**Proposition 5.**

## 4. Discussion

#### 4.1. Illustrative Examples

**Example 1.**

- 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 $(\{{g}_{j}I\mid {g}_{j}\in G\}\cup \left\{M\right\},\subseteq )$.

- 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 ${C}_{6}$.

**Example 2.**

- (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,$$\left\{\mathrm{Fever},\phantom{\rule{4pt}{0ex}}\mathrm{Aches},\phantom{\rule{4pt}{0ex}}\mathrm{Runny}\phantom{\rule{4pt}{0ex}}\mathrm{nose}\right\}\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}\left\{\mathrm{Fever},\phantom{\rule{4pt}{0ex}}\mathrm{Coughing},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{Runny}\phantom{\rule{4pt}{0ex}}\mathrm{nose}\right\}.$$

#### 4.2. A Comparative Analysis

**Example 3.**

#### 4.3. Potential Applications

- 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].

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Concept intents in Table 1.

**Figure 2.**Formal concepts in Table 1.

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 |

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

$\mathbf{Object}\phantom{\rule{3.33333pt}{0ex}}\mathit{g}$ | ${({\left\{\mathit{g}\right\}}^{\uparrow})}^{\downarrow}$ | ${\left[\mathit{g}\right]}_{{\mathit{R}}_{\mathit{M}}}$ |
---|---|---|

${g}_{1}$ | $\{{g}_{1},{g}_{3},{g}_{4}\}$ | $\left\{{g}_{1}\right\}$ |

${g}_{2}$ | $\{{g}_{2},{g}_{5}\}$ | $\{{g}_{2},{g}_{5}\}$ |

${g}_{3}$ | $\{{g}_{3},{g}_{4}\}$ | $\{{g}_{3},{g}_{4}\}$ |

${g}_{4}$ | $\{{g}_{3},{g}_{4}\}$ | $\{{g}_{3},{g}_{4}\}$ |

${g}_{5}$ | $\{{g}_{2},{g}_{5}\}$ | $\{{g}_{2},{g}_{5}\}$ |

${g}_{6}$ | $\left\{{g}_{6}\right\}$ | $\left\{{g}_{6}\right\}$ |

${g}_{7}$ | $\{{g}_{1},{g}_{3},{g}_{4},{g}_{6},{g}_{7}\}$ | $\left\{{g}_{7}\right\}$ |

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**MDPI and ACS Style**

Syau, Y.-R.; Lin, E.-B.; Liau, C.-J.
An Order-Theoretic Study on Formal Concept Analysis. *Axioms* **2023**, *12*, 1099.
https://doi.org/10.3390/axioms12121099

**AMA Style**

Syau Y-R, Lin E-B, Liau C-J.
An Order-Theoretic Study on Formal Concept Analysis. *Axioms*. 2023; 12(12):1099.
https://doi.org/10.3390/axioms12121099

**Chicago/Turabian Style**

Syau, Yu-Ru, En-Bing Lin, and Churn-Jung Liau.
2023. "An Order-Theoretic Study on Formal Concept Analysis" *Axioms* 12, no. 12: 1099.
https://doi.org/10.3390/axioms12121099