# Fuzzy Algebras of Concepts

^{*}

## Abstract

**:**

## 1. Introduction

- The creation of a gradation of the powerset lattice as an L-fuzzy lattice, that recovers the complete lattice of preconcepts for a specific case.
- The construction of an L-fuzzy algebra of preconcepts that preserves the subalgebra structure and recovers the algebra of protoconcepts for a specific case. We also extend this gradation to include both types of semiconcepts as well.
- The definition of an $L\times L\times L$-fuzzy algebra of preconcepts, that recovers the algebras of protoconcepts, semiconcepts, and formal concepts for different combinations of cuts.

## 2. Preliminaries

#### 2.1. Algebraic Conceptual Structures

**Remark**

**1.**

**Proposition**

**1**

**Definition**

**1.**

- 1.
- A pair $(A,B)\in {L}^{G}\times {L}^{M}$ is said to be a protoconcept if ${A}^{\uparrow \downarrow}={B}^{\downarrow}$ (or, equivalently, ${B}^{\downarrow \uparrow}={A}^{\uparrow}$).
- 2.
- A pair $(A,B)\in {L}^{G}\times {L}^{M}$ is said to be a ⊔-semiconcept if $A={B}^{\downarrow}$.
- 3.
- A pair $(A,B)\in {L}^{G}\times {L}^{M}$ is said to be a ⊓-semiconcept if ${A}^{\uparrow}=B$.

#### 2.2. L-Fuzzy Algebras and Lattices

**Definition**

**2**

**Definition**

**3**

**Proposition**

**2**

#### 2.3. Graded Subsethood of L-Fuzzy Sets

**graded subsethood**proposed by Bělohlávek [9], for L-fuzzy sets over a universe X, which is defined as follows:

**Definition**

**4**

**Theorem**

**1**

**.**Let $A,B$, C be L-fuzzy sets and ${\left\{{D}_{t}\right\}}_{t\in T}$ a family of L-fuzzy sets. Then:

- 1.
- If $A\subseteq B$, $S(B,C)\le S(A,C)$.
- 2.
- If $B\subseteq C$, $S(A,B)\le S(A,C)$.
- 3.
- $S(A,{\bigcap}_{t}{D}_{t})={\bigwedge}_{t}S(A,{D}_{t})$.
- 4.
- $S({\bigcup}_{t}{D}_{t},C)={\bigwedge}_{t}S({D}_{t},C)$.
- 5.
- $S(A,B)=1$ if, and only if, $A\subseteq B$.

## 3. Fuzzification of Conceptual Structures

#### 3.1. Fuzzy Preconcept Lattice

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Lemma**

**2.**

- 1.
- $\mathcal{V}\left(\underset{t}{\bigwedge}({A}_{t},{B}_{t})\right)\ge \underset{t}{\bigwedge}\mathcal{V}({A}_{t},{B}_{t})$.
- 2.
- $\mathcal{V}\left(\underset{t}{\bigvee}({A}_{t},{B}_{t})\right)\ge \underset{t}{\bigwedge}\mathcal{V}({A}_{t},{B}_{t})$.

**Proof.**

- 1.
- For the first item we have the following,$$\begin{array}{cc}\hfill \mathcal{V}\left(\underset{t}{\bigwedge}({A}_{t},{B}_{t})\right)& =\mathcal{V}\left(\bigcap _{t}{A}_{t},\bigcup _{t}{B}_{t}\right)=S\left(\bigcap _{t}{A}_{t},{\left(\bigcup _{t}{B}_{t}\right)}^{\downarrow}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \stackrel{\left(i\right)}{=}S\left(\bigcap _{t}{A}_{t},\bigcap _{t}{B}_{t}^{\downarrow}\right)\stackrel{\left(ii\right)}{=}\underset{t}{\bigwedge}S\left(\bigcap _{t}{A}_{t},{B}_{t}^{\downarrow}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \stackrel{\left(iii\right)}{\ge}\underset{t}{\bigwedge}S\left({A}_{t},{B}_{t}^{\downarrow}\right)=\underset{t}{\bigwedge}\mathcal{V}({A}_{t},{B}_{t}),\hfill \end{array}$$
- 2.
- For the second item we have,$$\begin{array}{cc}\hfill \mathcal{V}\left(\underset{t}{\bigvee}({A}_{t},{B}_{t})\right)& =\mathcal{V}\left(\bigcup _{t}{A}_{t},\bigcap _{t}{B}_{t}\right)=S\left(\bigcup _{t}{A}_{t},{\left(\bigcap _{t}{B}_{t}\right)}^{\downarrow}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \stackrel{\left(i\right)}{=}\underset{t}{\bigwedge}S\left({A}_{t},{\left(\bigcap _{t}{B}_{t}\right)}^{\downarrow}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \stackrel{\left(ii\right)}{\ge}\underset{t}{\bigwedge}S\left({A}_{t},{B}_{t}^{\downarrow}\right)=\underset{t}{\bigwedge}\mathcal{V}({A}_{t},{B}_{t}),\hfill \end{array}$$

**Theorem**

**3.**

**Proof.**

**Example**

**1.**

#### 3.2. Fuzzy Conceptual Algebras

**Lemma**

**3.**

- 1.
- $\underset{t}{{\displaystyle \sqcap}}\phantom{\rule{0.166667em}{0ex}}({A}_{t},{B}_{t})$ is a ⊓-semiconcept.
- 2.
- $\bigsqcup}_{t}\phantom{\rule{0.166667em}{0ex}}({A}_{t},{B}_{t})$ is a ⊔-semiconcept.

**Proof.**

#### 3.2.1. Fuzzy Algebra as a Measure of Protoconceptuality

**Lemma**

**4.**

- 1.
- $(A,B)$ is a protoconcept if, and only if, $\mathcal{P}(A,B)=1$.
- 2.
- $\mathcal{P}\left(\underset{t}{{\displaystyle \sqcap}}\phantom{\rule{0.166667em}{0ex}}({A}_{t},{B}_{t})\right)=\mathcal{P}\left({\displaystyle \bigsqcup}_{t}\phantom{\rule{0.166667em}{0ex}}({A}_{t},{B}_{t})\right)=1$.

**Proof.**

- 1.
- Let us suppose $(A,B)$ is a protoconcept. Then, it verifies, by Definition 1, ${B}^{\downarrow}={A}^{\downarrow \uparrow}$ hence, particularly, $\mathcal{P}(A,B)=S({B}^{\downarrow},{A}^{\downarrow \uparrow})=1$. Conversely, applying the rationale of the lines above this lemma, we have that if $\mathcal{P}(A,B)=1$ then $(A,B)$ is a protoconcept.
- 2.
- Note that both $\underset{t}{{\displaystyle \sqcap}}\phantom{\rule{0.166667em}{0ex}}({A}_{t},{B}_{t})$ and ${\displaystyle \bigsqcup}_{t}\phantom{\rule{0.166667em}{0ex}}({A}_{t},{B}_{t})$ are semiconcepts, by application of Lemma 3. Since every semiconcept is a protoconcept, by item 1 above, we have the desired result.

**Theorem**

**4.**

**Proof.**

**Example**

**2.**

#### 3.2.2. Fuzzy Algebra as a Measure of Semiconceptuality

**Lemma**

**5.**

- 1.
- $(A,B)$ is a ⊓-semiconcept if, and only if, ${\mathcal{H}}_{\sqcap}(A,B)=1$.
- 2.
- $(A,B)$ is a ⊔-semiconcept if, and only if, ${\mathcal{H}}_{\bigsqcup}(A,B)=1$.

**Proof.**

**Lemma**

**6.**

- 1.
- ${\mathcal{H}}_{\sqcap}\left(\underset{t}{{\displaystyle \sqcap}}\phantom{\rule{0.166667em}{0ex}}({A}_{t},{B}_{t})\right)\ge \underset{t}{\bigwedge}{\mathcal{H}}_{\sqcap}({A}_{t},{B}_{t})$.
- 2.
- ${\mathcal{H}}_{\sqcap}\left({\displaystyle \bigsqcup}_{t}\phantom{\rule{0.166667em}{0ex}}({A}_{t},{B}_{t})\right)\ge \underset{t}{\bigwedge}{\mathcal{H}}_{\sqcap}({A}_{t},{B}_{t})$.
- 3.
- ${\mathcal{H}}_{\bigsqcup}\left(\underset{t}{{\displaystyle \sqcap}}\phantom{\rule{0.166667em}{0ex}}({A}_{t},{B}_{t})\right)\ge \underset{t}{\bigwedge}{\mathcal{H}}_{\bigsqcup}({A}_{t},{B}_{t})$.
- 4.
- ${\mathcal{H}}_{\bigsqcup}\left({\displaystyle \bigsqcup}_{t}\phantom{\rule{0.166667em}{0ex}}({A}_{t},{B}_{t})\right)\ge \underset{t}{\bigwedge}{\mathcal{H}}_{\bigsqcup}({A}_{t},{B}_{t})$.

**Proof.**

- 1.
- Since, by Lemma 3, ${{\displaystyle \sqcap}}_{t}({A}_{t},{B}_{t})$ is a ⊓-semiconcept, then, applying Lemma 5, we have that ${\mathcal{H}}_{\sqcap}\left({{\displaystyle \sqcap}}_{t}({A}_{t},{B}_{t})\right)=1\ge {\bigwedge}_{t}{\mathcal{H}}_{\sqcap}({A}_{t},{B}_{t})$.
- 2.
- Let us consider a collection ${\left\{({A}_{t},{B}_{t})\right\}}_{t\in T}$ of at least two preconcepts. Then, the following chain of inequalities holds:$$\begin{array}{cc}\hfill {\mathcal{H}}_{\sqcap}\left({\displaystyle \bigsqcup}_{t}\phantom{\rule{0.166667em}{0ex}}({A}_{t},{B}_{t})\right)& ={\mathcal{H}}_{\sqcap}\left({\left(\bigcap _{t}{B}_{t}\right)}^{\downarrow},\bigcap _{t}{B}_{t}\right)=S\left({\left(\bigcap _{t}{B}_{t}\right)}^{\downarrow \uparrow},\bigcap _{t}{B}_{t}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \stackrel{\left(i\right)}{=}\underset{t}{\bigwedge}S\left({\left(\bigcap _{t}{B}_{t}\right)}^{\downarrow \uparrow},{B}_{t}\right)\stackrel{\left(ii\right)}{\ge}\underset{t}{\bigwedge}S\left({B}_{t}^{\downarrow \uparrow},{B}_{t}\right)=\underset{t}{\bigwedge}{\mathcal{H}}_{\sqcap}({A}_{t},{B}_{t}),\hfill \end{array}$$

**Theorem**

**5.**

**Proof.**

**Example**

**3.**

#### 3.2.3. Assembling Fuzzy Algebras

**Theorem**

**6.**

- 1.
- Its $(1,0,0)$-cut is the set of protoconcepts, $\mathfrak{V}\left(\mathbb{K}\right)$.
- 2.
- Its $(1,1,0)$-cut and its $(1,0,1)$-cut correspond to the sets of ⊓- and ⊔-semiconcepts, respectively.
- 3.
- Its $\mathbf{1}$-cut is the set of formal concepts.

**Proof.**

**Corollary**

**1.**

**Example**

**4.**

## 4. Conclusions

- A gradation of the powerset lattice, that preserves the complete sub-lattice property for all $\alpha $-cuts and recovers the complete lattice of preconcepts for the 1-cut.
- An L-fuzzy algebra of preconcepts, that preserves the subalgebra structure for all $\alpha $-cuts and recovers the algebra of protoconcepts for the 1-cut. Moreover, this gradation can be extended to include both types (⊓- and ⊔-) of semiconcepts as particular cases for different $\alpha $-cuts.
- Using the complete lattice $L\times L\times L$, induce an $L\times L\times L$-fuzzy algebra of preconcepts, which is able to recover the algebras of protoconcepts, ⊔-semiconcepts, ⊓-semiconcepts, and formal concepts for different combinations of $\alpha $-cuts: the $(1,0,0)$-cut is the algebra of protoconcepts, the $(0,1,0)$-cut is the set of ⊓-semiconcepts, the $(0,0,1)$-cut is the set of ⊔-semiconcepts, and the $(1,1,1)$-cut is the set of formal concepts.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary digital publishing institute |

DOAJ | Directory of open access journals |

FCA | Formal concept analysis |

## References

- Wille, R. Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts. Ordered Sets
**1982**, 83, 445–470. [Google Scholar] - Ganter, B.; Wille, R. Formal Concept Analysis—Mathematical Foundations; Springer: Berlin/Heidelberg, Germany, 1999. [Google Scholar]
- Luksch, P.; Wille, R. A Mathematical Model for Conceptual Knowledge Systems. In Classification, Data Analysis, and Knowledge Organization; Bock, H.H., Ihm, P., Eds.; Springer: Berlin/Heidelberg, Germany, 1991; pp. 156–162. [Google Scholar]
- Wagner, H. Begriff. In Handbuch Philosophischer Grundbegriffe; Krings, H., Baumgartner, H.M., Wild, C., Eds.; Kösel: München, Germany, 1973; pp. 191–209. [Google Scholar]
- Wille, R. Boolean Concept Logic. In Conceptual Structures: Logical, Linguistic, and Computational Issues, Proceedings of the 8th International Conference on Conceptual Structures, ICCS 2000, Darmstadt, Germany, 14–18 August 2000; Ganter, B., Mineau, G.W., Eds.; Springer: Berlin/Heidelberg, Germany, 2000; Volume 1867, pp. 317–331. [Google Scholar] [CrossRef]
- Wille, R. Formal Concept Analysis as Mathematical Theory of Concepts and Concept Hierarchies. In Formal Concept Analysis, Foundations and Applications; Ganter, B., Stumme, G., Wille, R., Eds.; Springer: Berlin/Heidelberg, Germany, 2005; Volume 3626, pp. 1–33. [Google Scholar] [CrossRef]
- Šostak, A.P.; Uljane, I.; Krastins, M. Gradation of Fuzzy Preconcept Lattices. Axioms
**2021**, 10, 41. [Google Scholar] [CrossRef] - Wille, R. Preconcept algebras and generalized double boolean algebras. In Concept Lattices, Proceedings of the Second International Conference on Formal Concept Analysis, ICFCA 2004, Sydney, Australia, 23–26 February 2004; Springer: Berlin/Heidelberg, Germany, 2004; pp. 1–13. [Google Scholar]
- Bělohlávek, R. Fuzzy Relational Systems; Springer: Berlin/Heidelberg, Germany, 2002. [Google Scholar]
- Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Piaget, J.; Herborth, F. Einführung in die genetische Erkenntnistheorie; Suhrkamp: Berlin, Germany, 1973. [Google Scholar]
- Stahl, J.; Wille, R. Preconcepts and set representation of contexts. In Classification as a Tool of Research; Gaul, W., Schader, M., Eds.; North–Holland: Amsterdam, The Netherlands, 1986; pp. 431–438. [Google Scholar]
- Ganter, B.; Kuznetsov, S.O. Stepwise construction of the Dedekind-MacNeille completion. In Conceptual Structures: Theory, Tools and Applications, Proceedings of the 6th International Conference on Conceptual Structures, ICCS’98 Montpellier, France, 10–12 August 1998; Springer: Berlin/Heidelberg, Germany, 1998; pp. 295–302. [Google Scholar]
- Burgmann, C.; Wille, R. The Basic Theorem on Preconcept Lattices. In Formal Concept Analysis, Proceedings of the 4th International Conference, ICFCA 2006, Dresden, Germany, 13–17 February 2006; Missaoui, R., Schmid, J., Eds.; Springer: Berlin/Heidelberg, Germany, 2006; Volume 3874, pp. 80–88. [Google Scholar] [CrossRef]
- Vormbrock, B.; Wille, R. Semiconcept and Protoconcept Algebras: The Basic Theorems. In Formal Concept Analysis, Foundations and Applications; Ganter, B., Stumme, G., Wille, R., Eds.; Springer: Berlin/Heidelberg, Germany, 2005; Volume 3626, pp. 34–48. [Google Scholar] [CrossRef]
- Howlader, P.; Banerjee, M. Kripke Contexts, Double Boolean Algebras with Operators and Corresponding Modal Systems. J. Log. Lang. Inf.
**2023**, 32, 117–146. [Google Scholar] [CrossRef] - Howlader, P.; Banerjee, M. A non-distributive logic for semiconcepts and its modal extension with semantics based on Kripke contexts. Int. J. Approx. Reason.
**2023**, 153, 115–143. [Google Scholar] [CrossRef] - Mao, H. Approximation operators for semiconcepts. J. Intell. Fuzzy Syst.
**2019**, 36, 3333–3343. [Google Scholar] [CrossRef] - Mao, H.; Cheng, Y.; Liu, X. Three-way preconcept and two forms of approximation operators. Soft Comput.
**2023**, 27, 855–865. [Google Scholar] [CrossRef] - Pawlak, Z. Rough sets. Int. J. Comput. Inf. Sci.
**1982**, 11, 341–356. [Google Scholar] [CrossRef] - Seselja, B.; Tepavcevic, A. Partially ordered and relational valued algebras and congruences. Rev. Res. Fac. Sci. Univ. Novi Sad. Math. Ser.
**1993**, 23, 273–287. [Google Scholar] - Tepavcevic, A.; Trajkovski, G. L-fuzzy lattices: An introduction. Fuzzy Sets Syst.
**2001**, 123, 209–216. [Google Scholar] [CrossRef]

**Figure 1.**L-fuzzy lattice and L-fuzzy algebras defined in this work. A double solid line stands for the L-fuzzy lattice, simple dashed lines represent L-fuzzy algebras, and a simple solid line represents the $\mathbb{O}$-fuzzy algebra whose cuts are $\mathfrak{P}\left(\mathbb{K}\right)$, $\mathfrak{P}{\left(\mathbb{K}\right)}_{\sqcap}$, $\mathfrak{P}{\left(\mathbb{K}\right)}_{\bigsqcup}$, and the set of formal concepts, $\mathbb{B}\left(\mathbb{K}\right)$.

${\mathit{m}}_{1}$ | ${\mathit{m}}_{2}$ | ${\mathit{m}}_{3}$ | |
---|---|---|---|

${g}_{1}$ | 1 | 0.2 | 0 |

${g}_{2}$ | 0.5 | 1 | 1 |

${g}_{3}$ | 0 | 0 | 0.7 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ojeda-Hernández, M.; López-Rodríguez, D.; Cordero, P.
Fuzzy Algebras of Concepts. *Axioms* **2023**, *12*, 324.
https://doi.org/10.3390/axioms12040324

**AMA Style**

Ojeda-Hernández M, López-Rodríguez D, Cordero P.
Fuzzy Algebras of Concepts. *Axioms*. 2023; 12(4):324.
https://doi.org/10.3390/axioms12040324

**Chicago/Turabian Style**

Ojeda-Hernández, Manuel, Domingo López-Rodríguez, and Pablo Cordero.
2023. "Fuzzy Algebras of Concepts" *Axioms* 12, no. 4: 324.
https://doi.org/10.3390/axioms12040324