# Gradation of Fuzzy Preconcept Lattices

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Lattices

#### 2.2. Quantales and Residuated Lattices

**Proposition**

**1.**

- (1)
- $\left(\right)open="("\; close=")">{\bigvee}_{i}{a}_{i}$for all$\left\{{a}_{i}\right|i\in I\}\subseteq \mathbb{L},$for all$b\in \mathbb{L}$;
- (2)
- $a\mapsto ({\bigwedge}_{i}{b}_{i})={\bigwedge}_{i}(a\mapsto {b}_{i})$for all$a\in \mathbb{L},$for all$\left\{{b}_{i}\right|i\in I\}\subseteq \mathbb{L}$;
- (3)
- ${1}_{\mathbb{L}}\mapsto a=a$for all$a\in \mathbb{L}$;
- (4)
- $a\mapsto b={1}_{\mathbb{L}}$whenever$a\le b;$
- (5)
- $a\ast (a\mapsto b)\le b$for all$a,b\in \mathbb{L}$;
- (6)
- $(a\mapsto b)\ast (b\mapsto c)\le a\mapsto c$for all$a,b,c\in \mathbb{L}$;
- (7)
- $a\mapsto b\le (a\ast c\mapsto b\ast c)$for all$a,b,c\in \mathbb{L};$
- (8)
- $a\ast b\le a\wedge bforanya,b\in \mathbb{L};$
- (9)
- $(a\ast b)\mapsto c=a\mapsto (b\mapsto c)$for any$a,b,c\in \mathbb{L}$.

#### 2.3. Fuzzy Sets and Fuzzy Relations

**Remark**

**1.**

#### 2.4. Measure of Inclusion of L-Fuzzy Sets

**Definition**

**1.**

**Proposition**

**2.**

- (1)
- $\left(\right)open="("\; close=")">{\bigvee}_{i}{A}_{i}$for all$\left\{{A}_{i}\right|i\in I\}\subseteq {L}^{X}$and for all$B\in {L}^{X};$
- (2)
- $A\hookrightarrow ({\bigwedge}_{i}{B}_{i})={\bigwedge}_{i}(A\hookrightarrow {B}_{i})$for all$A\in {L}^{X},$and for all$\left\{{B}_{i}\right|i\in I\}\subseteq {L}^{X};$
- (3)
- $A\hookrightarrow B={1}_{L}$whenever$A\le B$;
- (4)
- ${1}_{X}\hookrightarrow A={\bigwedge}_{x}A(x)$for all$A\in {L}^{X}$, where${1}_{X}:X\to L$is a constant function with the value${1}_{L}\in L$;
- (5)
- $(A\hookrightarrow B)\le (A\ast C\hookrightarrow B\ast C)$for all$A,B,C\in {L}^{X};$
- (6)
- $(A\hookrightarrow B)\ast (B\hookrightarrow C)\le (A\hookrightarrow C)$for all$A,B,C\in {L}^{X};$
- (7)
- $\left(\right)open="("\; close=")">{\bigwedge}_{i}{A}_{i}\ge {\bigwedge}_{i}({A}_{i}\hookrightarrow {B}_{i})$for all$\{{A}_{i}:i\in I\}$,$\{{B}_{i}:i\in I\}\subseteq {L}^{X}$;
- (8)
- $\left(\right)open="("\; close=")">{\bigvee}_{i}{A}_{i}\ge {\bigwedge}_{i}({A}_{i}\hookrightarrow {B}_{i})$for all$\{{A}_{i}:i\in I\}$,$\{{B}_{i}:i\in I\}\subseteq {L}^{X}.$

## 3. Preconcepts and Preconcept Lattices

**Definition**

**2.**

**Theorem**

**1.**

**Proof.**

## 4. Operators ${R}^{\uparrow}$ and ${R}^{\downarrow}$ on L-Powersets

**Definition**

**3.**

**Remark**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Example**

**1.**

**Proposition**

**6.**

**Proof.**

## 5. Concepts and Concept Lattices

**Definition**

**4.**

**Lemma**

**1.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**7.**

**Proof.**

**Remark**

**3.**

**Some topology-related comments**

- (1)
- $A\subseteq {c}_{X}(A)$(by Proposition 4), i.e., operator${c}_{X}:{L}^{X}\to {L}^{X}$is extensional,
- (2)
- ${A}_{1}\le {A}_{2}\u27f9{c}_{X}({A}_{1})\le {c}_{X}({A}_{2})$, i.e., operator${c}_{X}:{L}^{X}\to {L}^{X}$is isotone,
- (3)
- ${c}_{X}({c}_{X}(A))={({A}^{\uparrow \downarrow})}^{\uparrow \downarrow}={({A}^{\uparrow \downarrow \uparrow})}^{\downarrow}={({A}^{\downarrow})}^{\uparrow}={A}^{\uparrow \downarrow}={c}_{X}(A)$, i.e., operator${c}_{X}:{L}^{X}\to {L}^{X}$is idempotent.

**Theorem**

**2.**

**Proposition**

**8**

- 1.
- ${\u22cf}_{i\in I}{C}_{i}=\left(\right)open="("\; close=")">{\bigwedge}_{i\in I}{A}_{i},{\left(\right)}^{{\bigvee}_{i\in I}}\downarrow \uparrow $is its infimum in the partially ordered set$(\mathbb{C},\u2aaf)$.
- 2.
- ${\u22ce}_{i\in I}{C}_{i}=\left(\right)open="("\; close=")">{\left(\right)}^{{\bigvee}_{i\in I}}\uparrow \downarrow $is its supremum in the partially ordered set$(\mathbb{C},\u2aaf)$.

**Proof.**

**Corollary**

**2.**

- 1.
- ${\u22cf}_{i\in I}{C}_{i}=\left(\right)open="("\; close=")">{\bigwedge}_{i\in I}{A}_{i},{\left(\right)}^{{\bigwedge}_{i\in I}}\uparrow $is its infimum in the lattice$(\mathbb{C},\u2aaf)$.
- 2.
- ${\u22ce}_{i\in I}{C}_{i}=({\left(\right)}^{{\bigwedge}_{i\in I}}\downarrow ,{\bigwedge}_{i\in I}{B}_{i})$is its supremum in the lattice$(\mathbb{C},\u2aaf)$.

## 6. Conceptuality Degree of a Fuzzy Preconcept and $\mathcal{D}$-Graded Preconcept Lattices

#### 6.1. Degrees of Object and Attribute Based Contentments and the Degree of Conceptuality of a Fuzzy Preconcept

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

_{∗}) Operation ∗ has no zero divisors, i.e.,

- (${\u2020}_{A}^{B}$) ${\bigvee}_{y\in {B}^{c}}{\bigwedge}_{x\in A}R(x,y)=0$. In particular, this relation holds if $B=Y.$
- (${\u2020}_{B}^{A}$) ${\bigvee}_{y\in {A}^{c}}{\bigwedge}_{x\in B}R(x,y)=0$. In particular, this relation holds if $A=X.$
- $({\u2020}_{R})$ Thus, both conditions $({\u2020}_{{R}_{A}^{B}})$ and $({\u2020}_{{R}_{B}^{A}})$ are satisfied.
- $({\u2020}_{AB})$$A=X$ and $B=Y$.

**Example**

**2.**

- $\begin{array}{l}{A}^{\uparrow}\hookrightarrow B={\bigwedge}_{y\in Y}\left(\right)open="("\; close=")">{\bigwedge}_{x\in X}(A(x)\mapsto R(x,y))\mapsto B(y)& =\end{array}{\bigwedge}_{y\in {B}^{c}}\left(\right)open="("\; close=")">{\bigwedge}_{x\in X}(A(x)\mapsto R(x,y))\mapsto 0& ={\bigwedge}_{y\in {B}^{c}}\left(\right)open="("\; close=")">{\bigwedge}_{x\in A}(R(x,y)\mapsto 0)\\ ;$
- $\begin{array}{l}B\hookrightarrow {A}^{\uparrow}={\bigwedge}_{y\in Y}\left(\right)open="("\; close=")">B(y)\mapsto \left(\right)open="("\; close=")">{\bigwedge}_{x\in X}(A(x)\mapsto R(x,y))\\ =\end{array}$
- ${\mathcal{D}}^{\uparrow}(A,B)=\left(\right)open="("\; close=")">{\bigwedge}_{y\in {B}^{c}}\left(\right)open="("\; close=")">{\bigwedge}_{x\in A}(R(x,y)\mapsto 0)\wedge \left(\right)open="("\; close=")">{\bigwedge}_{x\in A,y\in B}R(x,y)$

- $A\hookrightarrow {B}^{\downarrow}={\bigwedge}_{x\in A,y\in B}R(x,y)$,${B}^{\downarrow}\hookrightarrow A={\bigwedge}_{x\in {A}^{c}}\left(\right)open="("\; close=")">{\bigwedge}_{y\in B}(R(x,y)\mapsto 0)$,
- ${\mathcal{D}}^{\downarrow}(A,B)=\left(\right)open="("\; close=")">{\bigwedge}_{x\in {A}^{c}}{\bigwedge}_{y\in B}(R(x,y)\mapsto 0)$and
- ${\mathcal{D}}^{\downarrow}(A,B)={\bigwedge}_{x\in A,y\in B}R(x,y)$in the case when either$({\u2020}_{\ast})$or$({\u2020}_{{R}_{AB}})$holds.

**Example**

**3.**

- $B\hookrightarrow {A}^{\uparrow}={\bigwedge}_{y\in Y}(B(y)\mapsto {A}^{\uparrow}(y))={\bigwedge}_{y\in B}{A}^{\uparrow}(y)={\bigwedge}_{y\in B,x\in {X}_{a}}(a\mapsto R(x,y));$
- ${A}^{\uparrow}\hookrightarrow B={\bigwedge}_{y\in {B}^{c}}\left(\right)open="("\; close=")">{\bigwedge}_{x\in {X}_{a}}(a\mapsto R(x,y))$; hence,
- ${\mathcal{D}}^{\uparrow}(A,B)=\left(\right)open="("\; close=")">{\bigwedge}_{y\in B,x\in {X}_{a}}(a\mapsto R(x,y))\mapsto 0$
- and${\mathcal{D}}^{\uparrow}(A,B)={\bigwedge}_{y\in B,x\in {X}_{a}}(a\mapsto R(x,y))$if condition$({\u2020}_{\ast})$or condition$({\u2020}_{{R}_{A}^{B}})$is satisfied.

- ${B}^{\downarrow}\hookrightarrow A=\left(\right)open="("\; close=")">{\bigwedge}_{x\in {X}_{a}}\left(\right)open="("\; close=")">{\bigwedge}_{y\in B}(R(x,y)\mapsto a)\wedge \left(\right)open="("\; close=")">{\bigwedge}_{x\in {X}_{a}^{c}}\left(\right)open="("\; close=")">{\bigwedge}_{y\in B}(R(x,y)\mapsto 0)$
- ${\mathcal{D}}^{\downarrow}(A,B)=\left(\right)open="("\; close=")">{\bigwedge}_{x\in {X}_{a}^{c}}\left(\right)open="("\; close=")">{\bigwedge}_{y\in B}(R(x,y)\mapsto 0)\wedge \left(\right)open="("\; close=")">{\bigwedge}_{x\in {X}_{a}}\left(\right)open="("\; close=")">{\bigwedge}_{y\in B}(R(x,y)\mapsto a)\wedge (a\mapsto {\bigwedge}_{y\in B}R(x,y))$

- $\mathcal{D}(A,B)={\bigwedge}_{x\in {X}_{a}}\left(\right)open="("\; close=")">{\bigwedge}_{y\in B}(R(x,y)\mapsto a)\wedge (a\mapsto {\bigwedge}_{y\in B}R(x,y))$

**Example**

**4.**

**Example**

**5.**

- (1)
- Łukasiewicz t-norm has zero divisors. Therefore, to simplify situation, we will consider the case when${X}_{a}=X$,$B=Y$, i.e., in the case when assumption$({\u2020}_{AB})$is satisfied. Then, from the above formulas, we have$${\mathcal{D}}^{\uparrow}(A,B)=\left(\right)open="("\; close=")">{\bigwedge}_{x\in X,y\in Y}(1-a+R(x,y))$$$${\mathcal{D}}^{\downarrow}(A,B)=\left(\right)open="("\; close=")">{\bigwedge}_{x\in X}(1-{\bigwedge}_{y\in Y}R(x,y)+a)$$$$\mathcal{D}(A,B)={\bigwedge}_{x\in X,y\in Y}(1-|a-R(x,y)|).$$
- (2)
- The product t-norm has no zero-divisors, i.e., it satisfies assumption$({\u2020}_{\ast})$. Hence, under this assumption, we can apply formulas obtained in Example 3 and have$$\mathcal{D}(A,B)=\left(\right)open="("\; close=")">{\bigwedge}_{x\in {X}_{a}}(a\mapsto {\bigwedge}_{y\in B}R(x,y))$$To describe$\mathcal{D}(A,B)$for the product t-norm in this situation, we denote${X}_{1}=\{x\in X|a<{\bigwedge}_{y\in B}R(x,y)\},$${X}_{2}=\{x\in X|a\ge {\bigwedge}_{y\in B}R(x,y)\}.$Then$$\mathcal{D}(A,B)=\left(\right)open="("\; close=")">\underset{x\in {X}_{1}}{\bigwedge}\frac{a}{{\bigwedge}_{y\in B}R(x,y)}.$$
- (3)
- The minimum t-norm has no zero-divisors, i.e., it satisfies assumption$({\u2020}_{\ast})$. Therefore, using formulas obtained in Example 3 and notations from the previous paragraph, we have$\mathcal{D}(A,B)=\left(\right)open="\{"\; close>\begin{array}{cc}{\bigwedge}_{x\in {X}_{2},y\in Y}R(x,y)\hfill & if{X}_{2}\ne \varnothing ;\hfill \\ a\hfill & otherwise\hfill \end{array}$

**Example**

**6.**

#### 6.2. $\mathcal{D}$-Graded Preconcept Lattices

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Theorem**

**5.**

## 7. Measure of Conceptuality of a Fuzzy Preconcept and $\mathcal{M}$-Graded Preconcept Lattices

#### 7.1. Conceptional Hull and Conceptional Kernel of a Fuzzy Preconcept

**Definition**

**9.**

- 1.
- $({A}^{0},{B}^{0})\u2aaf(A,B)$and
- 2.
- for every$(C,D)\in \mathbb{C}$such that$(C,D)\u2aaf(A,B)$it holds$({A}^{0},{B}^{0})\u2ab0(C,D)$.

**Definition**

**10.**

- 1.
- $(\overline{A},\overline{B})\u2ab0(A,B)$and
- 2.
- for every$(C,D)\in \mathbb{C}$such that$(C,D)\u2ab0(A,B)$it holds$(\overline{A},\overline{B})\u2aaf(C,D)$.

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

#### 7.2. Measure of Conceptuality of a Fuzzy Preconcept and $\mathcal{M}$-Graded Preconcept Lattices

**Definition**

**11.**

**Definition**

**12.**

**Definition**

**13.**

## 8. Appendix

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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A | B |
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40,000 | 30 |

60,000 | 35 |

80,000 | 40 |

90,000 | 45 |

100,000 | 50 |

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Šostak, A.; Uļjane, I.; Krastiņš, M.
Gradation of Fuzzy Preconcept Lattices. *Axioms* **2021**, *10*, 41.
https://doi.org/10.3390/axioms10010041

**AMA Style**

Šostak A, Uļjane I, Krastiņš M.
Gradation of Fuzzy Preconcept Lattices. *Axioms*. 2021; 10(1):41.
https://doi.org/10.3390/axioms10010041

**Chicago/Turabian Style**

Šostak, Alexander, Ingrīda Uļjane, and Māris Krastiņš.
2021. "Gradation of Fuzzy Preconcept Lattices" *Axioms* 10, no. 1: 41.
https://doi.org/10.3390/axioms10010041