# Relational Variants of Lattice-Valued F-Transforms

*Axioms*

**2020**,

*9*(1), 1; https://doi.org/10.3390/axioms9010001 (registering DOI)

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Residuated Lattices and $MV$-Algebras

- $(L,\vee ,\wedge ,{0}_{L}.{1}_{L})$ be a complete lattice,
- $(L,\otimes ,{1}_{L})$ be a commutative monoid,
- ⊗ is isotone in both arguments,
- → is a binary operation which is residuated with respect to ⊗, i.e.,$$\alpha \otimes \beta \le \gamma \phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{iff}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}\alpha \le \beta \to \gamma .$$

- (i)
- $(L,\otimes ,{1}_{L})$ is a commutative monoid,
- (ii)
- $(L,\oplus ,{0}_{L})$ is a commutative monoid,
- (iii)
- $\neg \neg x=x$, $\neg {0}_{L}={1}_{L}$,
- (iv)
- $x\oplus {1}_{L}={1}_{L}$, $x\oplus {0}_{L}=x$, $x\otimes {0}_{L}={0}_{L}$,
- (v)
- $x\oplus \neg x={1}_{L},x\otimes \neg x={0}_{L}$,
- (vi)
- $\neg (x\oplus y)=\neg x\otimes \neg y$, $\neg (x\otimes y)=\neg x\oplus \neg y$,
- (vii)
- $\neg (\neg x\oplus y)\oplus y=\neg (\neg y\oplus x)\oplus x$,

#### 2.2. Semirings and Semimodules

**Definition**

**1**

**.**A semiring $\mathcal{R}=(R,+,.,{0}_{R},{1}_{R})$ is an algebraic structure with the following properties:

- (i)
- $(R,+,{0}_{R})$ is a commutative monoid,
- (ii)
- $(R,.,{1}_{R})$ is a commutative monoid,
- (iii)
- $x.(y+z)=x.y+x.z$ holds for all $x,y,z\in R$,
- (iv)
- ${0}_{R}.x=x.{0}_{R}={0}_{R}$ holds for all $x\in R$.

**Definition**

**2**

**.**Let $\mathcal{R}=(R,+,.,{0}_{R},{1}_{R})$ be a semiring. A $\mathcal{R}$-semimodule is a commutative monoid $\mathcal{M}=(M,{+}_{M},{0}_{M})$ for which the external multiplication $R\times M\to M$, denoted by $rm$, is defined and which for all $r,{r}^{\prime}\in R$ and $m,{m}^{\prime}\in M$ satisfies the following equations:

- (o)
- $(r.{r}^{\prime})m=r\left({r}^{\prime}m\right)$,
- (ii)
- $r\left(m{+}_{M}{m}^{\prime}\right)=rm{+}_{M}r{m}^{\prime}$,
- (iii)
- $(r+{r}^{\prime})m=rm{+}_{M}{r}^{\prime}m$,
- (iv)
- ${1}_{R}m=m$, ${0}_{R}m=r{0}_{M}={0}_{M}$.

**Example**

**1**

**.**(1) Let $\mathcal{L}$ be a residuates lattice. Then the reduct ${\mathcal{L}}^{\vee}=(L,\vee ,\otimes ,{0}_{L},{1}_{L})$ is a commutative semiring,

**Example**

**2**

**.**(1) Let $X\ne \varnothing $, $\mathcal{L}$ be a residuated lattice and let ${\mathcal{L}}^{\vee}=(L,\vee ,\otimes ,{0}_{L},{1}_{L})$ be its semiring reduct. For all $f,g\in M={L}^{X}$ define

**Definition**

**3.**

- (i)
- $G\left(m{+}_{M}{m}^{\prime}\right)=G\left(m\right){+}_{N}G\left({n}^{\prime}\right)$, for all $m,{m}^{\prime}\in M$,
- (ii)
- $G\left(rm\right)=rG\left(m\right)$, for all $m\in M,r\in R$.

**Example**

**3.**

- 1.
- $\forall \{{s}_{i}:i\in I\}\subseteq M,\phantom{\rule{1.em}{0ex}}G\left({\bigvee}_{i\in I}^{M}{s}_{i}\right)={\bigvee}_{i\in I}^{N}G\left({s}_{i}\right)$,
- 2.
- $\forall s\in M,\alpha \in \mathcal{L},\phantom{\rule{1.em}{0ex}}G\left(\alpha {\otimes}_{M}s\right)=\alpha {\otimes}_{N}G\left(s\right)$.

**Example**

**4.**

- 1.
- $\forall \{{s}_{i}:i\in I\}\subseteq M,\phantom{\rule{1.em}{0ex}}G\left({\bigwedge}_{i\in I}^{M}{s}_{i}\right)={\bigwedge}_{i\in I}^{N}G\left({s}_{i}\right)$,
- 2.
- $\forall s\in M,\alpha \in \mathcal{L},\phantom{\rule{1.em}{0ex}}G\left(\alpha {\to}_{M}s\right)=\alpha {\to}_{N}G\left(s\right)$.

#### 2.3. Elements of the Category Theory

**Definition**

**4.**

- 1.
- For arbitrary objects $a,b,c\in Ob\left(\mathbf{K}\right)$, there exists a binary operation $\circ :Hom(a,b)\times Hom(b,c)\to Hom(a,c)$, called a composition of morphisms.
- 2.
- The composition ∘ is associative, i.e., if $f:a\to b,g:b\to c$ and $h:c\to d$, then $h\circ (g\circ f)=(h\circ g)\circ f$.
- 3.
- For every object $x\in Ob\left(\mathbf{K}\right)$ there exists a morphism ${1}_{x}:x\to x$, such that for arbitrary $f:x\to y$ and $g:y\to x$, $f\circ {1}_{x}=f$ and ${1}_{x}\circ g=g$ hold.

**Example**

**5.**

- 1.
- Objects are all ${\mathcal{L}}^{\vee}$-semimodules ${\mathcal{L}}^{X}$ from Example 2,
- 2.
- Morphisms are all ${\mathcal{L}}^{\vee}$-homomorphisms between ${\mathcal{L}}^{\vee}$-semimodules defined in Definition 4.

**Definition**

**5**

**.**

- 1.
- Let $\mathcal{L}$ be a complete residuated lattice. The category ${\mathbf{Hom}}_{\vee}$ is defined by
- (a)
- Objects are complete ${\mathcal{L}}^{\vee}$-semimodule homomorphism between ${\mathcal{L}}^{\vee}$-semimodules ${\mathcal{L}}^{X}$ and ${\mathcal{L}}^{Y}$, for arbitrary sets X and Y,
- (b)
- A morphism from an object $G:{\mathcal{L}}^{X}\to {\mathcal{L}}^{Y}$ to the object ${G}_{1}:{\mathcal{L}}^{{X}_{1}}\to {\mathcal{L}}^{{Y}_{1}}$ is a pair of maps $(f,\sigma ):G\to {G}_{1}$, such that $f:X\to {X}_{1}$ and $\sigma :Y\to {Y}_{1}$ are mappings, and holds$${G}_{1}\circ {f}^{\to}\ge {\sigma}^{\to}\circ G.$$
- (c)
- The composition of morphisms is point-wise.

- 2.
- Let $\mathcal{L}$ be a complete MV-algebra. The category ${\mathbf{Hom}}_{\wedge}$ is defined by
- (a)
- Objects are complete ${\mathcal{L}}^{\wedge}$-semimodule homomorphism between ${\mathcal{L}}^{\wedge}$-semimodules ${\mathcal{L}}^{X}$ and ${\mathcal{L}}^{Y}$, for arbitrary sets X and Y,
- (b)
- A morphism from an object $G:{\mathcal{L}}^{X}\to {\mathcal{L}}^{Y}$ to the object ${G}_{1}:{\mathcal{L}}^{{X}_{1}}\to {\mathcal{L}}^{{Y}_{1}}$ is a pair of maps $(f,\sigma ):G\to {G}_{1}$, such that $f:X\to {X}_{1}$ and $\sigma :Y\to {Y}_{1}$ are mappings, and holds$${\sigma}^{\leftarrow}\circ {G}_{1}\le G\circ {f}^{\leftarrow}.$$
- (c)
- The composition of morphisms is point-wise.

**Example**

**6.**

- 1.
- Let $\mathcal{L}$ be a complete residuated lattice. The category ${\mathbf{RHom}}_{\vee}$ is defined by
- (a)
- Objects are complete ${\mathcal{L}}^{\vee}$-semimodule homomorphism between ${\mathcal{L}}^{\vee}$-semimodules ${\mathcal{L}}^{X}$ and ${\mathcal{L}}^{Y}$, for arbitrary sets X and Y,
- (b)
- A morphism from an object $G:{\mathcal{L}}^{X}\to {\mathcal{L}}^{Y}$ to the object ${G}_{1}:{\mathcal{L}}^{{X}_{1}}\to {\mathcal{L}}^{{Y}_{1}}$ is a pair $(f,g):G\to {G}_{1}$, such that $f:X\times {X}_{1}\to \mathcal{L}$ and $g:Y\times {Y}_{1}\to \mathcal{L}$ are fuzzy relations, and in the diagramthe inequality ${G}_{1}\circ {f}^{\ast}\ge {g}^{\ast}\circ G$ holds.
- (c)
- The composition of morphisms is point-wise.

- 2.
- Let $\mathcal{L}$ be a complete MV-algebra. The category ${\mathbf{RHom}}_{\wedge}$ is defined by
- (a)
- Objects are complete ${\mathcal{L}}^{\wedge}$-semimodule homomorphism between ${\mathcal{L}}^{\wedge}$-semimodules ${\mathcal{L}}^{X}$ and ${\mathcal{L}}^{Y}$, for arbitrary sets X and Y,
- (b)
- A morphism from an object $G:{\mathcal{L}}^{X}\to {\mathcal{L}}^{Y}$ to the object ${G}_{1}:{\mathcal{L}}^{{X}_{1}}\to {\mathcal{L}}^{{Y}_{1}}$ is a pair $(f,g):G\to {G}_{1}$, such that $f:X\times {X}_{1}\to \mathcal{L}$ and $g:Y\times {Y}_{1}\to \mathcal{L}$ are fuzzy relations, and in the following diagramthe inequality $G\circ {f}_{\ast}\ge {g}_{\ast}\circ {G}_{1}$ holds.
- (c)
- The composition of morphisms is point-wise.

**Definition**

**6.**

- 1.
- associates to each object x in $\mathbf{C}$ an object $F\left(x\right)$ in $\mathbf{D}$,
- 2.
- associates to each morphism $f:x\to y$ in $\mathbf{C}$ a morphism $F\left(f\right):F\left(x\right)\to F\left(y\right)$ in $\mathbf{D}$ such that the following two conditions hold:
- (a)
- $F\left({1}_{x}\right)={1}_{F\left(x\right)}$, for arbitrary object x in $\mathbf{C}$,
- (b)
- $F(g\circ f)=F\left(g\right)\circ F\left(f\right)$, for arbitrary $f:a\to b,g:b\to c$.

**Example**

**7.**

- 1.
- For arbitrary $X\in Ob\left(\mathbf{Set}\right)$, $F\left(X\right)={\mathcal{L}}^{X}\in {\mathbf{Mod}}^{\vee}$. Elements of $F\left(X\right)$ are called $\mathcal{L}$-fuzzy sets in X.
- 2.
- For a morphism $f:X\to Y$ in $\mathbf{Set}$, $F\left(f\right):F\left(X\right)\to F\left(Y\right)$ is defined by $F\left(s\right)={s}^{\to}$, for arbitrary $s\in F\left(X\right)$.

## 3. Categories of Spaces with Fuzzy Partitions

**Definition**

**7.**

**Definition**

**8.**

- 1.
- Fuzzy partitions $(X,\mathcal{A})$ as objects,
- 2.
- Morphisms $(g,\sigma ):(X,\mathcal{A})\to (Y,\mathcal{B})$ such that
- $g:X\to Y$ is a map,
- $\sigma :\left|\mathcal{A}\right|\to \left|\mathcal{B}\right|$ is a map such that$$\forall \lambda \in \left|\mathcal{A}\right|,x\in X\phantom{\rule{1.em}{0ex}}{A}_{\lambda}\left(x\right)\le {B}_{\sigma \left(\lambda \right)}\left(g\left(x\right)\right).$$

- 3.
- The composition of morphisms in $\mathbf{SpaceFP}$ is defined by $(h,\tau )\circ (g,\sigma )=(h\circ g,\tau \circ \sigma )$.

**Definition**

**9.**

- 1.
- Ob(
**RSpaceFP**)=Ob(**SpaceFP**), - 2.
- $(f,g):(X,\mathcal{A})\to (Y,\mathcal{B})$ is a morphism, if
- (a)
- $f:X\times Y\to \mathcal{L}$ is a $\mathcal{L}$-fuzzy relation,
- (b)
- $g:\left|\mathcal{A}\right|\times \left|\mathcal{B}\right|\to \mathcal{L}$ is a $\mathcal{L}$-fuzzy relation,
- (c)
- For all $x\in X,y\in Y,\alpha \in \left|\mathcal{A}\right|,\beta \in \left|\mathcal{B}\right|$,$$\begin{array}{c}\underset{\alpha \in \left|\mathcal{A}\right|}{\bigvee}{A}_{\alpha}\left(x\right)\otimes g(\alpha ,\beta )\le \underset{y\in Y}{\bigvee}{B}_{\beta}\left(y\right)\otimes f(x,y),\\ \underset{x\in X}{\bigvee}{A}_{\alpha}\left(x\right)\otimes f(x,y)\le \underset{\beta \in \left|\mathcal{B}\right|}{\bigvee}S(\alpha ,\beta )\otimes {B}_{\beta}\left(y\right).\end{array}$$

- 3.
- A composition of morphisms $(f,g):(X,\mathcal{A})\to (Y,\mathcal{B})$ and $({f}_{1},{g}_{1}):(Y,\mathcal{B})\to (Z,\mathcal{C})$ is a morphism $({f}_{1}\circ f,{g}_{1}\circ g):(X,\mathcal{A})\to (Z,\mathcal{C})$, where ∘ is a standard composition of $\mathcal{L}$-fuzzy relations.

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

- $I\left(X\right)=(X,\left\{{\chi}_{\left\{x\right\}}^{X}:x\in X\right\})$, where ${\chi}_{\left\{x\right\}}^{X}:X\to \mathcal{L}$ is the characteristic function of a set $\left\{x\right\}$ in a set X,
- If $f:X\to Y$ is a morphism in $\mathbf{SpaceFP}$, then $I\left(f\right)=(f,f)$.

**Definition**

**10.**

- 1.
- Objects $(X,Y,R)$, where $X,Y$ are sets and $R:X\times Y\to \mathcal{L}$ is an $\mathcal{L}$-fuzzy relation.
- 2.
- Morphisms are pair of maps $(f,\sigma ):(X,Y,R)\to ({X}^{\prime},{Y}^{\prime},{R}^{\prime})$, where $f:X\to {X}^{\prime}$ and $\sigma :Y\to {Y}^{\prime}$ are maps, such that $R(x,y)\le {R}^{\prime}(f\left(x\right),\sigma \left(y\right))$, for all $x\in X,y\in Y$,
- 3.
- The composition of morphisms is defined by $(g,\tau )\circ (f,\sigma )=(g\circ f,\tau \circ \sigma )$.

**Definition**

**11.**

- 1.
- Objects of $\mathbf{RRel}$ are the same as in the category $\mathbf{Rel}$,
- 2.
- $(f,g):(X,Y,R)\to ({X}_{1},{Y}_{1},{R}_{1})$ is a morphism, if
- (a)
- $f:X\times {X}_{1}\to \mathcal{L}$ is a fuzzy relation,
- (b)
- $g:Y\times {Y}_{1}\to \mathcal{L}$ is a fuzzy relation,
- (c)
- $g\circ R\le {R}_{1}\circ f$ and $f\circ {R}^{-1}\le {S}^{-1}\circ g$ hold, where ∘ is a standard composition of fuzzy relations.

- 3.
- A composition of morphisms $(f,g):(X,Y,R)\to ({X}_{1},{Y}_{1},{R}_{1})$ and $({f}_{1},{g}_{1}):({X}_{1},{Y}_{1},{R}_{1})\to ({X}_{2},{Y}_{2},{R}_{2})$ is a morphism $(f\circ {f}_{1},g\circ {g}_{1}):(X,Y,R)\to ({X}_{2},{Y}_{2},{R}_{2})$.

**Proposition**

**4.**

**Proof.**

- $H(X,Y,R)=(X,\mathcal{A})$, where $\mathcal{A}=\{{A}_{y}:y\in Y\}$ and ${A}_{y}:X\to \mathcal{L}$ is defined by ${A}_{y}\left(x\right)=R(x,y)$.
- $H(f,g)=(f,g)$.

- ${H}^{-1}(X,\mathcal{A})=(X,|\mathcal{A}|,R)$, where $R:X\times \left|\mathcal{A}\right|\to \mathcal{L}$ is defined by $R(x,\alpha )={A}_{\alpha}\left(x\right)$.
- ${H}^{-1}(f,\sigma )=(f,\sigma )$.

## 4. Categories of F-Transforms

**Definition**

**12.**

- 1.
- A function ${F}_{X,\mathcal{A}}^{\uparrow}:{\mathcal{L}}^{X}\to {\mathcal{L}}^{\left|\mathcal{A}\right|}$ is called the upper F-transform defined by a space with a fuzzy partition, if$$\forall f\in {\mathcal{L}}^{X},y\in \left|\mathcal{A}\right|,\phantom{\rule{1.em}{0ex}}{F}_{X,\mathcal{A}}^{\uparrow}\left(f\right)\left(y\right)=\underset{x\in X}{\bigvee}f\left(x\right)\otimes {A}_{y}\left(x\right).$$
- 2.
- A function ${F}_{X,\mathcal{A}}^{\downarrow}:{\mathcal{L}}^{X}\to {\mathcal{L}}^{\left|\mathcal{A}\right|}$ is called the lower F-transform defined by a space with a fuzzy partition, if$$\forall f\in {\mathcal{L}}^{X},y\in \left|\mathcal{A}\right|,\phantom{\rule{1.em}{0ex}}{F}_{X,\mathcal{A}}^{\downarrow}\left(f\right)\left(y\right)=\underset{x\in X}{\bigwedge}({A}_{y}\left(x\right)\to f\left(x\right))$$

**Definition**

**13.**

- 1.
- Objects are upper F-transform maps ${F}_{X,\mathcal{A}}^{\uparrow}:{\mathcal{L}}^{X}\to {\mathcal{L}}^{\left|\mathcal{A}\right|}$, for arbitrary space with a fuzzy partition $(X,\mathcal{A})$,
- 2.
- A morphism from ${F}_{X,\mathcal{A}}^{\uparrow}$ to ${F}_{Y,\mathcal{B}}^{\uparrow}$ is a pair o maps $(f,\sigma )$, such that $f:X\to Y$ and $\sigma :\left|\mathcal{A}\right|\to \left|\mathcal{B}\right|$ are maps, such that$${F}_{Y,\mathcal{B}}^{\uparrow}.{f}^{\to}\ge {\sigma}^{\to}.{F}_{X,\mathcal{A}}^{\uparrow}.$$
- 3.
- Composition of morphisms is a component-wise composition of maps.

**Definition**

**14.**

- 1.
- Objects of ${\mathbf{RFTrans}}^{\uparrow}$ are the same as for the category ${\mathbf{FTrans}}^{\uparrow}$,
- 2.
- A morphism from ${F}_{X,\mathcal{A}}^{\uparrow}$ to ${F}_{Y,\mathcal{B}}^{\uparrow}$ is a pair $(R,S)$ of $\mathcal{L}$-fuzzy relations such that $R:X\times Y\to \mathcal{L}$, $S:\left|\mathcal{A}\right|\times \left|\mathcal{B}\right|\to \mathcal{L}$, such that$${F}_{Y,\mathcal{B}}^{\uparrow}.{R}^{\ast}\ge {S}^{\ast}.{F}_{X,\mathcal{A}}^{\uparrow}.$$
- 3.
- Composition of morphisms is a component-wise composition of $\mathcal{L}$-fuzzy relations.

**Definition**

**15.**

- 1.
- Objects are lower F-transform maps ${F}_{X,\mathcal{A}}^{\downarrow}:{\mathcal{L}}^{X}\to {\mathcal{L}}^{\left|\mathcal{A}\right|}$, for arbitrary space with a fuzzy partition $(X,\mathcal{A})$,
- 2.
- A morphism from ${F}_{X,\mathcal{A}}^{\downarrow}$ to ${F}_{Y,\mathcal{B}}^{\downarrow}$ is a pair o maps $(f,\sigma )$, such that $f:X\to Y$ and $\sigma :\left|\mathcal{A}\right|\to \left|\mathcal{B}\right|$ are maps, such that$${\sigma}^{\leftarrow}.{F}_{Y,\mathcal{B}}^{\downarrow}\le {F}_{X,\mathcal{A}}^{\downarrow}.{f}^{\leftarrow}.$$
- 3.
- Composition of morphisms is a component-wise composition of maps.

**Definition**

**16.**

- 1.
- Objects of ${\mathbf{RFTrans}}^{\downarrow}$ are the same as for the category ${\mathbf{FTrans}}^{\downarrow}$,
- 2.
- A morphism from ${F}_{X,\mathcal{A}}^{\downarrow}$ to ${F}_{Y,\mathcal{B}}^{\downarrow}$ is a pair $(R,S)$ of $\mathcal{L}$-fuzzy relations such that $R:X\times Y\to \mathcal{L}$, $S:\left|\mathcal{A}\right|\times \left|\mathcal{B}\right|\to \mathcal{L}$, such that$${S}_{\ast}.{F}_{Y,\mathcal{B}}^{\downarrow}\le {F}_{X,\mathcal{A}}^{\downarrow}.{R}_{\ast}.$$
- 3.
- Composition of morphisms is a component-wise composition of $\mathcal{L}$-fuzzy relations.

**Proposition**

**5.**

**Proof.**

- ${\mathcal{F}}^{\uparrow}(X,\mathcal{A})={F}_{X,\mathcal{A}}^{\uparrow}$,
- ${\mathcal{F}}^{\uparrow}(f,\sigma )=(f,\sigma )$.

- ${\mathcal{F}}^{\downarrow}(X,\mathcal{A})={F}_{X,\mathcal{A}}^{\downarrow}$,
- ${\mathcal{F}}^{\downarrow}(f,\sigma )=(f,\sigma )$.

**Theorem**

**1**

**.**

- 1.
- Let $\mathcal{L}$ be a complete residuated lattice. Then the categories $\mathbf{SpaceFP}$ and ${\mathbf{Hom}}_{\vee}$ are isomorphic.
- 2.
- Let $\mathcal{L}$ be a complete MV-algebra. Then the categories $\mathbf{SpaceFP}$ and ${\mathbf{Hom}}_{\wedge}$ are also isomorphic.

**Theorem**

**2.**

- 1.
- Let $\mathcal{L}$ be a complete residuated lattice. Then the categories ${\mathbf{RFTrans}}^{\uparrow}$ and ${\mathbf{RHom}}_{\vee}$ are identical.
- 2.
- Let $\mathcal{L}$ be a complete MV-algebra. Then the categories ${\mathbf{RFTrans}}^{\downarrow}$ and ${\mathbf{RHom}}_{\wedge}$ are identical.

**Proof.**

**Example**

**8.**

## 5. Conclusions and Prospective Results

## Funding

## Conflicts of Interest

## References

- Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Goguen, J.A. L-fuzzy sets. J. Math. Anal. Appl.
**1967**, 18, 145–174. [Google Scholar] [CrossRef] [Green Version] - Goguen, J.A. Categories of L-sets. Bull. Am. Math. Soc.
**1969**, 75, 622–637. [Google Scholar] [CrossRef] - Eytan, M. Fuzzy sets, a topos-theoretical point of view. Fuzzy Sets Syst.
**1981**, 5, 47–67. [Google Scholar] [CrossRef] - Barr, M. Fuzzy sets and topos theory. Canad. Math. Bull.
**1986**, 29, 501–508. [Google Scholar] [CrossRef] - Höhle, U.; Stout, L.N. Foundations of Fuzzy Sets. Fuzzy Sets Syst.
**1991**, 40, 257–296. [Google Scholar] [CrossRef] - Higgs, D. A Category Approach to Boolean-Valued Set Theory; Preprint; University of Waterloo: Waterloo, ON, Canada, 1973. [Google Scholar]
- Higgs, D. Injectivity in the topos of complete Heyting valued algebra sets. Canad. J. Math.
**1984**, 36, 550–568. [Google Scholar] [CrossRef] - Wyler, O. Fuzzy Logic and Categories of Fuzzy Sets. In Non-Classical Logics and Their Applications to Fuzzy Subsets; Höhle, U., Klement, E.P., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1995; pp. 235–268. [Google Scholar]
- Blanc, G. Préfaisceaux et Ensembles Totalement Floux; University of Aix-marseille: Marseille, France, 1981. [Google Scholar]
- Höhle, U. M-Valued Sets and Sheaves Over Integral Commutative CL-Monoids. In Applications of Category Theory to Fuzzy Subsets; Rodabaugh, S.E., Klement, E.P., Höhle, U., Eds.; Springer Science+Business Media: Berlin, Germany, 1992; pp. 33–72. [Google Scholar]
- Höhle, U. GL-Quantales: Q-Valued Sets and Their Singletons. Studia Logica
**1998**, 61, 123–148. [Google Scholar] [CrossRef] - Höhle, U.; Kubiak, T. Quantale Sets and Their Singleton Monad; Preprint; Elsevier North-Holland, Inc.: Amsterdam, The Netherlands, 2008. [Google Scholar]
- Solovyov, S.A. On the Category Set(JCPos). Fuzzy Sets Syst.
**2006**, 157, 459–465. [Google Scholar] [CrossRef] - Solovyov, S.A. Categories of Lattice-Valued Sets as Categories of Arrows. Fuzzy Sets Syst.
**2006**, 157, 843–854. [Google Scholar] [CrossRef] - Perfilieva, I.; Singh, A.P.; Tiwari, S.P. On the relationship among F-transform, fuzzy rough set and fuzzy topology. In Proceedings of the IFSA-EUSFLAT, Asturias, Spain, 30 Jun–3 July 2015; Atlantis Press: Amsterdam, The Netherlands, 2015; pp. 1324–1330. [Google Scholar] [CrossRef]
- Wang, C.Y. Fuzzy rough sets based on generalized residuated lattices. Inf. Sci.
**2013**, 248, 31–49. [Google Scholar] [CrossRef] - Wu, W.Z.; Leung, Y.; Mi, J.S. On characterization of ($\mathcal{J}$, $\mathcal{T}$)-fuzzy rough approximation operators. Fuzzy Sets Syst.
**2005**, 154, 76–102. [Google Scholar] [CrossRef] - Di Martino, F.; Loia, V.; Perfilieva, I.; Sessa, S. An image coding/decoding method based on direct and inverse fuzzy tranforms. Int. J. Approx. Reason.
**2008**, 48, 110–131. [Google Scholar] [CrossRef] [Green Version] - Di Martino, F.; Loia, V.; Sessa, S. A segmentation method for images compressed by fuzzy transforms. Fuzzy Sets Syst.
**2010**, 161, 56–74. [Google Scholar] [CrossRef] - Di Martino, F.; Sessa, S. Compression and decompression of images with discrete fuzzy transforms. Inf. Sci.
**2007**, 177, 2349–2362. [Google Scholar] [CrossRef] - Di Martino, F.; Loia, V.; Sessa, S. Fuzzy transforms method and attribute dependency in data analysis. Inf. Sci.
**2010**, 180, 493–505. [Google Scholar] [CrossRef] - Di Martino, F.; Loia, V.; Sessa, S. Fuzzy transforms method in prediction data analysis. Fuzzy Sets Syst.
**2011**, 180, 146–163. [Google Scholar] [CrossRef] - Perfilieva, I.; Novak, V.; Dvořak, A. Fuzzy transforms in the analysis of data. Int. J. Approx. Reason.
**2008**, 48, 36–46. [Google Scholar] [CrossRef] [Green Version] - Perfilieva, I. Fuzzy transforms and their applications to image compression. In Fuzzy Logic and Applications; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2006; pp. 19–31. [Google Scholar]
- Stefanini, L. F-transform with parametric generalized fuzzy partitions. Fuzzy Sets Syst.
**2011**, 180, 98–120. [Google Scholar] [CrossRef] - Khastan, A.; Perfilieva, I.; Alijani, Z. A new fuzzy approximation method to Cauchy problem by fuzzy transform. Fuzzy Sets Syst.
**2016**, 288, 75–95. [Google Scholar] [CrossRef] - Tomasiello, S. An alternative use of fuzzy transform with application to a class of delay differential equations. Int. J. Comput. Math.
**2017**, 94, 1719–1726. [Google Scholar] [CrossRef] - Štěpnička, M.; Valašek, R. Numerical solution of partial differential equations with the help of fuzzy transform. In Proceedings of the FUZZ-IEEE 2005, Reno, NV, USA, 25 May 2005; pp. 1104–1109. [Google Scholar]
- Perfilieva, I. Fuzzy transforms: A challange to conventional transform. In Advances in Image and Electron Physics; Hawkes, P.W., Ed.; Elsevies Academic Press: San Diego, CA, USA, 2007; Volume 147, pp. 137–196. [Google Scholar] [CrossRef]
- Perfilieva, I. Fuzzy transforms: Theory and applications. Fuzzy Sets Syst.
**2006**, 157, 993–1023. [Google Scholar] [CrossRef] - Perfilieva, I.; Valasek, R. Fuzzy transforms in removing noise. Adv. Soft Comput.
**2005**, 2, 221–230. [Google Scholar] [CrossRef] - Perfilieva, I.; Kreinovich, V. Fuzzy transforms of higher order approximate derivatives: A theorem. Fuzzy Sets Syst.
**2011**, 180, 55–68. [Google Scholar] [CrossRef] [Green Version] - Perfilieva, I.; De Baets, B. Fuzzy transforms of monotone functions with application to image compression. Inf. Sci.
**2010**, 180, 3304–3315. [Google Scholar] [CrossRef] - Novák, V.; Perfilijeva, I.; Močkoř, J. Mathematical Principles of Fuzzy Logic; Kluwer Academic Publishers: Boston, MA, USA; Dordrecht, The Netherlands; London, UK, 1991. [Google Scholar]
- Berstel, J.; Perrin, D. Theory of Codes; Academic Press: Orlando, FL, USA, 1985. [Google Scholar]
- Golan, J.S. Semirings and Their Applications; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1999. [Google Scholar]
- Di Nola, A.; Gerla, B. Algebras of Łukasiewicz logic and their semiring reducts. Contemp. Math.
**2005**, 377, 131–144. [Google Scholar] - Di Nola, A.; Lettieri, A.; Perfilieva, I.; Novák, V. Algebraic analysis of fuzzy systems. Fuzzy Sets Syst.
**2007**, 158, 1–22. [Google Scholar] [CrossRef] [Green Version] - Mac Lane, S. Categories for the Working Mathematician; Graduate text in Mathematics; Springer Science+Business Media: New York, NY, USA, 1971; Volume 5. [Google Scholar]
- Močkoř, J. F-transforms and Semimodule Homomorphisms. Soft Comput.
**2019**, 23, 7603–7619. [Google Scholar] [CrossRef] - Rodabaugh, S.E. Powerset operator based foundation for point-set lattice theoretic (poslat) fuzzy set theories and topologies. Quaest. Math.
**1997**, 20, 463–530. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Močkoř, J.
Relational Variants of Lattice-Valued F-Transforms. *Axioms* **2020**, *9*, 1.
https://doi.org/10.3390/axioms9010001

**AMA Style**

Močkoř J.
Relational Variants of Lattice-Valued F-Transforms. *Axioms*. 2020; 9(1):1.
https://doi.org/10.3390/axioms9010001

**Chicago/Turabian Style**

Močkoř, Jiří.
2020. "Relational Variants of Lattice-Valued F-Transforms" *Axioms* 9, no. 1: 1.
https://doi.org/10.3390/axioms9010001