# Semiring-Valued Fuzzy Sets and F-Transform

## Abstract

**:**

## 1. Introduction

- To introduce the notion of $\mathcal{R}$-fuzzy sets and to show that a significant part of $\mathcal{L}$-fuzzy type structures, where $\mathcal{L}$ is the complete $MV$-algebra, can be transformed into $\mathcal{R}$-fuzzy sets,
- To show that for $\mathcal{R}$-fuzzy sets it is possible to define analogies of concepts and transformations with analogous properties known from the classical $\mathcal{L}$-fuzzy sets,
- To show that these new concepts and transformations for $\mathcal{R}$-fuzzy sets can be transformed back into concepts and transformations of the original $\mathcal{L}$-fuzzy type structures.

## 2. Methods and Basic Structures

- (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)
- $\neg (\neg x\oplus y)\oplus y=\neg (\neg y\oplus x)\oplus x$.

**Definition**

**1**

- 1.
- $\mathcal{A}$ is called a fuzzy partition, if $\{core\left({A}_{y}\right):y\in Y\}$ is a partition of X, where $core\left(A\right)=\{x\in X:A\left(x\right)={1}_{L}\}$, i.e., ${\bigcup}_{y\in Y}core\left({A}_{y}\right)=X$, $core\left({A}_{y}\right)\cap core\left({A}_{z}\right)=\varnothing $, if $y\ne z$.
- 2.
- A mapping ${F}_{X,\mathcal{A}}:{\mathcal{L}}^{X}\to {\mathcal{L}}^{Y}$ is called the upper F-transform based on $\mathcal{A}$, if for $s\in {\mathcal{L}}^{X},y\in Y$, ${F}_{X,\mathcal{A}}\left(s\right)\left(y\right)={\bigvee}_{x\in X}s\left(x\right)\otimes {A}_{y}\left(x\right)$.
- 3.
- A mapping ${G}_{X,\mathcal{A}}:{\mathcal{L}}^{Y}\to {\mathcal{L}}^{X}$ is called the inverse upper F-transform based on $\mathcal{A}$, if for $g\in {\mathcal{L}}^{Y},x\in X$, ${G}_{X,\mathcal{A}}\left(g\right)\left(x\right)={\bigwedge}_{y\in Y}\neg {A}_{y}\left(x\right)\oplus g\left(y\right)$.

**Definition**

**2**

- (i)
- $(R,+,{0}_{R})$ is an idempotent commutative monoid,
- (ii)
- $(R,\times ,{1}_{R})$ is a commutative monoid,
- (iii)
- $x\times (y+z)=x\times y+x\times z$ holds for all $x,y,z\in R$,
- (iv)
- ${0}_{R}\times x={0}_{R}$ holds for all $x\in R$.
- (v)
- $(R,\le )$ is a partially pre-ordered (or ordered) set such that for all $a,b,c\in R$ the following hold$$\begin{array}{c}a\le b\Rightarrow a{+}_{R}c\le b{+}_{R}c,\phantom{\rule{1.em}{0ex}}a{\times}_{R}c\le b{\times}_{R}c,\\ a\ge {0}_{R}.\end{array}$$

**Example**

**1**

- (1)
- Let $\mathcal{L}$ be a residuates lattice. Then the reduct ${\mathcal{L}}^{\vee}=(L,\le ,\vee ,\otimes ,{0}_{L},{1}_{L})$ is the $po$-semiring.
- (2)
- Let $\mathcal{L}$ be a $MV$-algebra. Then the reduct ${\mathcal{L}}^{\wedge}=(L,\le ,\wedge ,\oplus ,{1}_{L},{0}_{L})$ is the $po$-semiring.

**Definition**

**3**

- 1.
- $(M,\u229e,0)$ is a commutative monoid,
- 2.
- $\star :M\times R\to M$ is a mapping (called an external multiplication),
- 3.
- $r,{r}^{\prime}\in R,m\in M,\phantom{\rule{3.33333pt}{0ex}}\left(r{\times}_{R}{r}^{\prime}\right)\star m=r\star ({r}^{\prime}\star m)$,
- 4.
- $r\in R,m,{m}^{\prime}\in M,\phantom{\rule{3.33333pt}{0ex}}r\star (m\u229e{m}^{\prime})=r\star m\u229er\star {m}^{\prime}$,
- 5.
- $r,{r}^{\prime}\in R,m\in M,\phantom{\rule{3.33333pt}{0ex}}\left(r{+}_{R}{r}^{\prime}\right)\star m=r\star m\u229e{r}^{\prime}\star m$,
- 6.
- ${1}_{R}\star m=m$, ${0}_{R}\star m=r\star 0=0$,
- 7.
- $m,n,p\in M,m\le n\Rightarrow m\u229ep\le n\u229ep$,
- 8.
- $m,n\in M,r\in R$, $r{\ge}_{R}{0}_{R},m\le n\Rightarrow r\star m\le r\star n$,
- 9.
- $r,s\in R,m\in M$, $r{\le}_{R}s\Rightarrow r\star m\le s\star m$,

**Example**

**2**

- (1)
- Let $X\ne \varnothing $, $\mathcal{L}$ be a complete residuated lattice and let ${\mathcal{L}}^{\vee}=(L,\le ,\vee ,\otimes ,{0}_{L},{1}_{L})$ be the $po$-semiring from Example 1. For all $f,g\in M={L}^{X}$ define$$\begin{array}{c}f\le g\iff \forall x\in X,f\left(x\right)\le g\left(x\right)\phantom{\rule{4.pt}{0ex}}in\phantom{\rule{4.pt}{0ex}}{\mathcal{L}}^{\vee},\\ \left(f{\oplus}_{M}g\right)\left(x\right)=f\left(x\right)\vee g\left(x\right),\\ p{\star}_{1}f\left(x\right)=p\otimes f\left(x\right),\\ {0}_{M}\in M,\phantom{\rule{1.em}{0ex}}{0}_{M}\left(x\right)={0}_{L},\phantom{\rule{1.em}{0ex}}x\in X,p\in L.\end{array}$$Then ${\mathcal{L}}^{X}=(M,\le ,{\oplus}_{M},{\star}_{1},{0}_{M})$ is the complete $po$-${\mathcal{L}}^{\vee}$-semimodule.
- (2)
- Let $X\ne \varnothing $, $\mathcal{L}$ be a complete $MV$-algebra and let ${\mathcal{L}}^{\wedge}=(L,\le ,\wedge ,\oplus ,{1}_{L},{0}_{L})$ be the $po$-semiring from Example 1. For all $f,g\in M={L}^{X}$ define$$\begin{array}{c}f\le g\iff \forall x\in X,f\left(x\right)\ge g\left(x\right)\phantom{\rule{4.pt}{0ex}}in\phantom{\rule{4.pt}{0ex}}{\mathcal{L}}^{\wedge},\\ \left(f{\oplus}_{M}g\right)\left(x\right)=f\left(x\right)\wedge g\left(x\right),\\ p{\star}_{2}f\left(x\right)=p\oplus f\left(x\right),\\ {0}_{M}\in M,\phantom{\rule{1.em}{0ex}}{0}_{M}\left(x\right)={1}_{L},\phantom{\rule{1.em}{0ex}}x\in X,p\in L.\end{array}$$Then ${\mathcal{L}}_{X}=(M,\le ,{\oplus}_{M},{\star}_{2},{0}_{M})$ is the complete $po$-${\mathcal{L}}^{\wedge}$-semimodule.

**Definition**

**4.**

- 1.
- A $po$-semiring homomorphism $\Phi :\mathcal{R}\to \mathcal{S}$ is a mapping $\Phi :R\to S$ such that
- (a)
- Φ is a homomorphism of semirings,
- (b)
- Φ is order-preserving.

- 2.
- A $\mathcal{R}$-semimodule homomorphism $\Psi :\mathcal{M}\to \mathcal{N}$ is a mapping $\Psi :M\to N$ such that
- (a)
- $\Psi :\mathcal{M}\to \mathcal{N}$ is an order preserving homomorphism of monoids,
- (b)
- $\Psi \left(r{\star}_{M}m\right)=r{\star}_{N}\Psi \left(m\right)$, for all $m\in M,r\in R$,

**Example**

**3.**

## 3. Results

#### 3.1. $\mathcal{R}$-Valued Fuzzy Sets

**Definition**

**5.**

- 1.
- Φ is an order-preserving isomorphism of $po$-semirings,
- 2.
- Φ is self-inverse, i.e., $\Phi .\Phi =i{d}_{R}$,
- 3.
- $\forall a,b\in \mathcal{R}$, $a\le b\iff a{\ge}^{*}b$,
- 4.
- $\forall a,b,c\in \mathcal{R},\phantom{\rule{3.33333pt}{0ex}}a{\times}^{*}(b+c)=\left(a{\times}^{*}b\right)+\left(a{\times}^{*}c\right)$,
- 5.
- $\forall a,b,c\in \mathcal{R},\phantom{\rule{3.33333pt}{0ex}}a+\left(b{+}^{*}c\right)=(a+b){+}^{*}(a+c)$,

**Remark**

**1.**

- 1.
- In the rest of the paper, if $(\mathcal{R},{\mathcal{R}}^{*})$ will be the adjoint pair of semirings with the adjoint isomorphism Φ, then $\mathcal{R}$ and ${\mathcal{R}}^{*}$ are supposed to be complete $po$-semirings with the same operations as in Definition 5.
- 2.
- It should be observed that the following statements dual to statements from Definition 5 also holds:
- 4’.
- $\forall a,b,c\in \mathcal{R},\phantom{\rule{3.33333pt}{0ex}}a\times \left(b{+}^{*}c\right)=(a\times b){+}^{*}(a\times c)$,
- 5’.
- $\forall a,b,c\in \mathcal{R},\phantom{\rule{3.33333pt}{0ex}}a{+}^{*}(b+c)=\left(a{+}^{*}b\right)+\left(a{+}^{*}c\right)$.

**Example**

**4.**

**Example**

**5.**

- 1.
- The partially pre-ordered semiring ${\mathcal{R}}_{1}=({R}_{1},{\le}_{1},{+}_{1},{\times}_{1},{0}_{1},{1}_{1})$ is defined by
- (a)
- ${R}_{1}=\{A:A\subseteq L\}={2}^{L}$,
- (b)
- $A,B\in {R}_{1},\phantom{\rule{3.33333pt}{0ex}}A{+}_{1}B=A\cup B$,
- (c)
- $A,B\in {R}_{1},\phantom{\rule{3.33333pt}{0ex}}A{\times}_{1}B:=A\otimes B=\{a\otimes b:a\in A,b\in B\},\phantom{\rule{3.33333pt}{0ex}}A{\times}_{1}\varnothing =\varnothing $,
- (d)
- ${0}_{1}=\varnothing ,{1}_{1}=\left\{{1}_{L}\right\}$,
- (e)
- $A,B\in {R}_{1}$, we set$$A{\le}_{1}B\iff (\forall \alpha \in A)(\exists \beta \in B)\alpha \le \beta .$$

- 2.
- The partially pre-ordered semiring ${\mathcal{R}}_{1}^{*}=({R}_{1},{\le}_{1}^{*},{+}_{1}^{*},{\times}_{1}^{*},{0}_{1}^{*},{1}_{1}^{*})$ is defined by
- (a)
- $A,B\in {R}_{1}$, $A{+}_{1}^{*}B=A\cap B$,
- (b)
- For $A,B\in {R}_{1}$, $A,B\ne L$, $A{\times}_{1}^{*}B:=A\oplus B=\{a\oplus b:a\in A,b\in B\}$, $A{\times}_{1}^{*}L=L{\times}_{1}^{*}L=L$,
- (c)
- ${0}_{1}^{*}=L,\phantom{\rule{3.33333pt}{0ex}}{1}_{1}^{*}=\left\{{0}_{L}\right\}$.
- (d)
- $A{\le}_{1}^{*}B\iff B{\le}_{1}A$.

**Example**

**6.**

- 1.
- The $po$-semiring ${\mathcal{R}}_{2}=({R}_{2},{\le}_{2},{+}_{2},{\times}_{2},{0}_{2},{1}_{2})$ is defined by
- (a)
- ${R}_{2}=\{(\alpha ,\beta )\in {L}^{2}:\neg \alpha \ge \beta \}\subseteq {L}^{2}$,
- (b)
- $(\alpha ,\beta ){+}_{2}({\alpha}_{1},{\beta}_{1}):=(\alpha \vee {\alpha}_{1},\beta \wedge {\beta}_{1})$,
- (c)
- $(\alpha ,\beta ){\times}_{2}({\alpha}_{1},{\beta}_{1}):=(\alpha \otimes {\alpha}_{1},\beta \oplus {\beta}_{1})$,
- (d)
- ${0}_{2}=({0}_{L},{1}_{L}),\phantom{\rule{3.33333pt}{0ex}}{1}_{2}=({1}_{L},{0}_{L})$,
- (e)
- $(\alpha ,\beta ){\le}_{2}({\alpha}^{\prime},{\beta}^{\prime})\iff \alpha \le {\alpha}^{\prime},\beta \ge {\beta}^{\prime}.$

- 2.
- The $po$-semiring ${\mathcal{R}}_{2}^{*}=({R}_{2},{\le}_{2}^{*},{+}_{2}^{*},{\times}_{2}^{*},{0}_{2}^{*},{1}_{2}^{*})$ is defined by
- (a)
- $(\alpha ,\beta ){+}_{2}^{*}({\alpha}_{1},{\beta}_{1}):=(\alpha \wedge {\alpha}_{1},\beta \vee {\beta}_{1})$,
- (b)
- $(\alpha ,\beta ){\times}_{2}^{*}({\alpha}_{1},{\beta}_{1}):=(\alpha \oplus {\alpha}_{1},\beta \otimes {\beta}_{1})$,
- (c)
- ${0}_{2}^{*}=({1}_{L},{0}_{L}),\phantom{\rule{3.33333pt}{0ex}}{1}_{2}^{*}=({0}_{L},{1}_{L})$,
- (d)
- $(\alpha ,\beta ){\le}_{2}^{*}({\alpha}^{\prime},{\beta}^{\prime})\iff (\alpha ,\beta ){\ge}_{2}({\alpha}^{\prime},{\beta}^{\prime})$.

**Example**

**7.**

- 1.
- Let K be the fixed set of criteria. The $po$-semiring ${\mathcal{R}}_{3}=({R}_{3},{\le}_{3},{+}_{3},{\times}_{3},{0}_{3},{1}_{3})$ is defined by
- (a)
- ${R}_{3}=\{(E,\psi ):E\subseteq K,\psi \in {L}^{K}\}\subseteq {L}^{K}$, where $(E,\psi )\in {L}^{K}$ is defined by$$k\in K,\phantom{\rule{1.em}{0ex}}(E,\psi )\left(k\right)=\left\{\begin{array}{cc}\psi \left(k\right),\hfill & k\in E,\hfill \\ {0}_{L},\hfill & k\notin E\hfill \end{array}\right..$$
- (b)
- $(E,\phi ),(F,\psi )\in {R}_{3}$, $(E,\phi ){+}_{3}(F,\psi ):=(E\cap F,\phi \vee \psi )$, where $\phi \vee \psi $ is the supremum in ${L}^{K}$,
- (c)
- $(E,\phi ),(F,\psi )\in {R}_{3}$, $(E,\phi ){\times}_{3}(F,\psi )=(E\cap F,\phi \times \psi )$, where $\phi \times \psi \in {L}^{K}$ is defined by $\phi \times \psi \left(k\right)=\phi \left(k\right)\otimes \psi \left(k\right)$,
- (d)
- ${0}_{3}=(K,{\underline{0}}_{L})$, ${1}_{3}=(K,{\underline{1}}_{L})$, where $\underline{\alpha}\left(k\right)=\alpha $ for arbitrary $k\in K$, $\alpha \in L$,
- (e)
- $(E,\phi ){\le}_{3}(F,\psi )\iff (E,\phi )\left(k\right)\le (F,\psi )\left(k\right),\forall k\in E\cap F$.

- 2.
- The $po$-semiring ${\mathcal{R}}_{3}^{*}=({R}_{3},{\le}_{3}^{*},{+}_{3}^{*},{\times}_{3}^{*},{0}_{3}^{*},{1}_{3}^{*})$ is defined by
- (a)
- $(E,\phi ),(F,\psi )\in {R}_{3}$, $(E,\phi ){+}_{3}^{*}(F,\psi ):=(E\cap F,\phi \wedge \psi )$, where $\phi \wedge \psi $ is the infimum in ${L}^{K}$,
- (b)
- $(E,\phi ),(F,\psi )\in {R}_{3}$, $(E,\phi ){\times}_{3}^{*}(F,\psi )=(E\cap F,\phi \oplus \psi )$, where ⊕ in ${L}^{K}$ is defined component-wise.
- (c)
- ${0}_{3}^{*}=(K,{\underline{1}}_{L})$, ${1}_{3}^{*}=(K,{\underline{0}}_{L})$, where $\underline{\alpha}\left(k\right)=\alpha $ for arbitrary $k\in K$, $\alpha \in L$,
- (d)
- $(E,\phi ){\le}_{3}^{*}(F,\psi )\iff (E,\phi ){\ge}_{3}(F,\psi )$.

**Example**

**8.**

- 1.
- The $po$-semiring ${\mathcal{R}}_{4}=({R}_{4},{\le}_{4},{+}_{4},{\times}_{4},{0}_{4},{1}_{0})$ is defined by
- (a)
- ${R}_{4}={L}^{3}$,
- (b)
- $(\alpha ,\beta ,\gamma ){+}_{4}({\alpha}_{1},{\beta}_{1},{\gamma}_{1}):=(\alpha \vee {\alpha}_{1},\beta \wedge {\beta}_{1},\gamma \wedge {\gamma}_{1})$,
- (c)
- $(\alpha ,\beta ,\gamma ){\times}_{2}({\alpha}_{1},{\beta}_{1},{\gamma}_{1}):=(\alpha \otimes {\alpha}_{1},\beta \wedge {\beta}_{1},\gamma \oplus {\gamma}_{1})$,
- (d)
- ${0}_{4}=({0}_{L},{1}_{L},{1}_{L}),\phantom{\rule{1.em}{0ex}}{1}_{4}=({1}_{L},{1}_{L},{0}_{L})$,
- (e)
- $(\alpha ,\beta ,\gamma ){\le}_{4}({\alpha}^{\prime},{\beta}^{\prime},{\gamma}^{\prime})\iff \alpha \le {\alpha}^{\prime},\beta \ge {\beta}^{\prime},\gamma \ge {\gamma}^{\prime}.$

- 2.
- The $po$-semiring ${\mathcal{R}}_{4}^{*}=({R}_{4},{\le}_{4}^{*},{+}_{4}^{*},{\times}_{4}^{*},{0}_{4}^{*},{1}_{4}^{*})$ is defined by
- (a)
- $(\alpha ,\beta ,\gamma ){+}_{4}^{*}({\alpha}_{1},{\beta}_{1},{\gamma}_{1}):=(\alpha \wedge {\alpha}_{1},\beta \vee {\beta}_{1},\gamma \vee {\gamma}_{1})$,
- (b)
- $(\alpha ,\beta ,\gamma ){\times}_{4}^{*}({\alpha}_{1},{\beta}_{1},{\gamma}_{1}):=(\alpha \oplus {\alpha}_{1},\beta \vee {\beta}_{1},\gamma \otimes {\gamma}_{1})$,
- (c)
- ${0}_{4}^{*}=({1}_{L},{0}_{L},{0}_{L}),\phantom{\rule{1.em}{0ex}}{1}_{4}^{*}=({0}_{L},{0}_{L},{1}_{L})$,
- (d)
- $(\alpha ,\beta ,\gamma ){\le}_{4}^{*}({\alpha}^{\prime},{\beta}^{\prime},{\gamma}^{\prime})\iff (\alpha ,\beta ,\gamma ){\ge}_{4}({\alpha}^{\prime},{\beta}^{\prime},{\gamma}^{\prime})$.

**Example**

**9.**

**Definition**

**6.**

- 1.
- A mapping $s:X\to \mathcal{R}$ is called the $\mathcal{R}$-fuzzy set in X. For $x\in X$, $s\left(x\right)\in \mathcal{R}$ is called the $\mathcal{R}$-membership value of s in x.
- 2.
- The operations with $\mathcal{R}$-fuzzy sets $s,t$ in X and elements $a\in \mathcal{R}$ are defined by
- (a)
- The intersection $s\sqcap t$ is defined by $(s\sqcap t)\left(x\right)=s\left(x\right){+}^{*}t\left(x\right)$, $x\in X$,
- (b)
- The union $s\bigsqcup t$ is defined by $(s\bigsqcup t)\left(x\right)=s\left(x\right)+t\left(x\right)$, $x\in X$,
- (c)
- The complement $\neg s$ is defined by $\neg s\left(x\right)=\Phi \left(s\right(x\left)\right)$,
- (d)
- The external multiplication ☆ by elements from $\mathcal{R}$ is defined by$(a\star s)\left(x\right)=a\times s\left(x\right)$,
- (e)
- The pre-order relation ⊆ between $\mathcal{R}$-fuzzy sets is defined by$$s\subseteq t\iff (\forall x\in X)s\left(x\right)\le t\left(x\right).$$

- 3.
- For arbitrary $x\in X$, by ${\eta}_{X}\left(x\right)$ we denote the $\mathcal{R}$-fuzzy set in X such that$$y\in X,\phantom{\rule{1.em}{0ex}}{\eta}_{X}\left(x\right)\left(y\right)=\left\{\begin{array}{cc}{1}_{\mathcal{R}},\hfill & x=y,\hfill \\ {0}_{\mathcal{R}},\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

**Lemma**

**1.**

- 1.
- $s\sqcap s=s,s\bigsqcup s=s$,
- 2.
- $s\sqcap t\subseteq s$,
- 3.
- $s\sqcap (t\bigsqcup w)=(s\sqcap t)\bigsqcup (s\sqcap w)$,
- 4.
- $s\bigsqcup (t\sqcap w)=(s\bigsqcup t)\sqcap (s\bigsqcup w)$,
- 5.
- $a\star (t\bigsqcup w)=(a\star t)\bigsqcup (a\star w)$,
- 6.
- $a\star (t\sqcap w)=(a\star t)\sqcap (a\star w)$,
- 7.
- $\neg (s\bigsqcup t)=\neg s\sqcap \neg t$, $\neg (s\sqcap t)=\neg s\bigsqcup \neg t$,
- 8.
- $s\subseteq t\Rightarrow s\bigsqcup w\subseteq t\bigsqcup w,\phantom{\rule{1.em}{0ex}}s\sqcap w\subseteq t\sqcap w$,

**Proof.**

**Definition**

**7.**

- 1.
- For arbitrary $f,g\in {R}^{X},x\in X$, $(f\u229eg)\left(x\right)=f\left(x\right){+}_{R}g\left(x\right)$,
- 2.
- For arbitrary $f\in {R}^{X},r\in R,x\in X$, $(r\star f)\left(x\right)=r{\times}_{R}f\left(x\right)$,
- 3.
- ${0}_{X}\left(x\right)={0}_{R}$,
- 4.
- $f\le g\iff \forall x\in X,f\left(x\right){\le}_{R}g\left(x\right).$

**Remark**

**2.**

**Definition**

**8**

- 1.
- A hesitant $\mathcal{L}$-fuzzy set in a set X is a mapping $h:X\to {2}^{L}$, i.e., for $x\in X$, $h\left(x\right)\subseteq L$. By $H\left(X\right)$ we denote the set of all hesitant fuzzy sets in X.
- 2.
- An intuitionistic $\mathcal{L}$-fuzzy set in a set X is a pair $(u,v)$ of $\mathcal{L}$-fuzzy sets on X, such that $\neg u\ge v$. By $J\left(X\right)$ we denote the set of all intuitionistic fuzzy sets in X.
- 3.
- A neutrosophic $\mathcal{L}$-fuzzy set in a set X is a triple $(u,v,w)$ of $\mathcal{L}$-fuzzy sets on X, called a truth membership function u, an indeterminancy membership function v and a falsity membership function w. By $N\left(X\right)$ we denote the set of all neutrosophis fuzzy sets in X.
- 4.
- Let K be the fixed set of criteria. A pair $(E,s)$ is called an $\mathcal{L}$-fuzzy soft set in the set X, if $\varnothing \ne E\subseteq K$ and $s:E\to {\mathcal{L}}^{X}$. By $S\left(X\right)$ we denote the set of all fuzzy soft sets in X.

**Remark**

**3.**

**Notation**

**1.**

**Proposition**

**1.**

- 1.
- The algebraic structure $\left(H\right(X),\cap ,\cup ,\neg ,\circ )$ of all hesitant $\mathcal{L}$-fuzzy sets in X is transformable to ${\mathcal{R}}_{1}$-fuzzy sets.
- 2.
- The algebraic structure $\left(J\right(X),\cap ,\cup ,\neg ,\circ )$ of all intuitionistic $\mathcal{L}$-fuzzy sets is transformable to ${\mathcal{R}}_{2}$-fuzzy sets.
- 3.
- The algebraic structure $\left(S\right(X),\cap ,\cup ,\neg ,\circ )$ of all $\mathcal{L}$-fuzzy soft sets in X is transformable ${\mathcal{R}}_{3}$-fuzzy sets.
- 4.
- The algebraic structure $\left(N\right(X),\cap ,\cup ,\neg ,\circ )$ of all neutrosophic $\mathcal{L}$-fuzzy sets in X is transformable to ${\mathcal{R}}_{4}$-fuzzy sets.

**Proof.**

- (1)
- Any hesitant $\mathcal{L}$-fuzzy set is a mapping $X\to {2}^{L}$ and it follows that $H\left(X\right)={\mathcal{R}}_{1}^{X}$. The isomorphism of operations follows directly from the definitions of operations in ${\mathcal{R}}_{1}^{X}$ and in $H\left(X\right)$.
- (2)
- Any intuitionistic $\mathcal{L}$-fuzzy set is a mapping $X\to \{(\alpha ,\beta )\in {L}^{2}:\neg \alpha \ge \beta \}$ and it follows that $J\left(X\right)={\mathcal{R}}_{2}^{X}$. The isomorphism of operations follows directly from the definitions of operations in ${\mathcal{R}}_{2}^{X}$ and in $J\left(X\right)$.
- (3)
- According to Remark 3, any $\mathcal{L}$-fuzzy soft set $(E,s)\in S\left(X\right)$ is a mapping $(E,s):X\to {L}^{K}$, where $E\subseteq K$, $s:K\to {L}^{X}$ and$$k\in K,x\in X,\phantom{\rule{1.em}{0ex}}(E,s)\left(x\right)\left(k\right)=\left\{\begin{array}{cc}s\left(k\right)\left(x\right),\hfill & k\in E,\hfill \\ {0}_{L},\hfill & k\notin E\hfill \end{array}\right..$$We define the mapping $\Lambda :S\left(X\right)\to {\mathcal{R}}_{3}^{X}$ such that$$\begin{array}{c}(E,s)\in S\left(X\right),\phantom{\rule{1.em}{0ex}}\Lambda (E,s):X\to {\mathcal{R}}_{3},\\ x\in X,\phantom{\rule{1.em}{0ex}}\Lambda (E,s)\left(x\right):=(E,{s}_{x})\in {\mathcal{R}}_{3},\\ {s}_{x}:K\to L,\phantom{\rule{1.em}{0ex}}{s}_{x}\left(k\right):=s\left(k\right)\left(x\right)\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}k\in K,\\ (E,{s}_{x}):K\to L,\phantom{\rule{1.em}{0ex}}(E,{s}_{x})\left(k\right):=\left\{\begin{array}{cc}s\left(k\right)\left(x\right),\hfill & k\in E,\hfill \\ {0}_{L},\hfill & k\notin E.\hfill \end{array}\right.\end{array}$$We prove that $\Lambda \left(S\left(X\right)\right)={\mathcal{R}}_{3}^{X}$. Because $\Lambda (E,s)\in {\mathcal{R}}_{3}^{X}$, we have $\Lambda \left(S\left(X\right)\right)\subseteq {\mathcal{R}}_{3}^{X}$ and we need to prove only the inverse inclusion. Let $g\in {\mathcal{R}}_{3}^{X}$, $g\left(x\right)=({E}_{x},{\psi}_{x})\in {\mathcal{R}}_{3}$ for $x\in X$. Let the element $(E,s)\in S\left(X\right)$ be defined by$$\begin{array}{c}E=\bigcup _{x\in X}{E}_{x},\\ s:K\to {L}^{X},\phantom{\rule{1.em}{0ex}}s\left(k\right)\left(x\right)=\left\{\begin{array}{cc}{\psi}_{x}\left(k\right),\hfill & k\in {E}_{x},\hfill \\ {0}_{L},\hfill & k\notin {E}_{x}.\hfill \end{array}\right.\end{array}$$For $x\in X$ we have $\Lambda (E,s)\left(x\right)=(E,{s}_{x})$ and we show that $\Lambda (E,s)\left(x\right)$ and $g\left(x\right)$ are equal as mappings $K\to L$. For $k\in K$ we have$$\begin{array}{c}(E,{s}_{x})\left(k\right)=\left\{\begin{array}{cc}s\left(k\right)\left(x\right),\hfill & k\in E,\hfill \\ {0}_{L},\hfill & k\notin E\hfill \end{array}\right.=\\ \left\{\begin{array}{cc}{\psi}_{x}\left(k\right),\hfill & k\in {E}_{x},\hfill \\ {0}_{L},\hfill & k\in E\setminus {E}_{x}\hfill \\ {0}_{L},\hfill & k\notin E\hfill \end{array}\right.=\left\{\begin{array}{cc}{\psi}_{x}\left(k\right),\hfill & k\in {E}_{x},\hfill \\ {0}_{L},\hfill & k\notin {E}_{x}\hfill \end{array}\right.=g\left(x\right)\left(k\right).\end{array}$$Therefore, $\Lambda (E,s)=g$ and $\Lambda \left(S\left(X\right)\right)={\mathcal{R}}_{3}^{X}$. It is easy to see that $\Lambda $ is the injective map. The isomorphism of operations in ${\mathcal{R}}_{3}^{X}$ and in $S\left(X\right)$ follows directly from definitions of these operations in ${\mathcal{R}}_{3}^{X}$ and $S\left(X\right)$. Therefore, the algebraic structure $S\left(X\right)$ is isomorphic to the algebraic structure ${\mathcal{R}}_{3}^{X}$.
- (4)
- Any neutrosphic $\mathcal{L}$-fuzzy set is a mapping $X\to {L}^{3}$ and it follows that $N\left(X\right)={\mathcal{R}}_{4}^{X}$. The isomorphism of operations follows directly from the definitions of operations in ${\mathcal{R}}_{4}^{X}$ and in $N\left(X\right)$.

**Lemma**

**2.**

- 1.
- The set $B=\{{\eta}_{X}\left(x\right):x\in X\}$ is the $\mathcal{R}$-base of the free $\mathcal{R}$-semimodule ${\mathcal{R}}^{X}$.
- 2.
- The set $\neg B=\{\neg {\eta}_{X}\left(x\right):x\in X\}$ is the ${\mathcal{R}}^{*}$-base of the free ${\mathcal{R}}^{*}$-semimodule ${\left({\mathcal{R}}^{*}\right)}^{X}$, where $\neg \left({\eta}_{X}\left(x\right)\right)\left(z\right)=\Phi \left({\eta}_{X}\left(x\right)\left(z\right)\right)$, for $z\in X$.

**Proof.**

- (1)
- We show firstly that the following identity holds for arbitrary $f\in {\mathcal{R}}^{X}$:$$f=\underset{x\in X}{\overset{{\mathcal{R}}^{X}}{\u229e}}f\left(x\right)\star {\eta}_{X}\left(x\right).$$In fact, according to Definition 7, for $a\in X$ we obtain$$\left(\underset{x\in X}{\overset{{\mathcal{R}}^{X}}{\u229e}}f\right(x)\star {\eta}_{X}(x\left)\right)\left(a\right)=\sum _{x\in X}^{\mathcal{R}}f\left(x\right){\times}_{R}{\eta}_{X}\left(x\right)\left(a\right)=f\left(a\right)\times {}_{R}{1}_{R}=f\left(a\right)$$$$\underset{x\in X}{\overset{{\mathcal{R}}^{X}}{\u229e}}{r}_{x}\star \eta X\left(x\right)=0$$$${0}_{R}=\underset{x\in X}{\overset{{\mathcal{R}}^{X}}{\u229e}}{r}_{x}\star \eta X\left(x\right)\left)\right(a)=\sum _{x\in X}^{\mathcal{R}}f(x\left){\times}_{R}{\eta}_{X}\right(x\left)\right(a)={r}_{a}.$$Therefore, B is the $\mathcal{R}$-base of ${\mathcal{R}}^{X}$.
- (2)
- We show that for arbitrary $f\in {\left({\mathcal{R}}^{*}\right)}^{X}$ we have$$f=\underset{x\in X}{\overset{{\left({\mathcal{R}}^{*}\right)}^{X}}{\u229e}}f\left(x\right)\star *(\neg \eta X(x\left)\right).$$In fact, according to Definition 5 and Definition 7, for $a\in X$ we have$$\left(\underset{x\in X}{\overset{{\left({\mathcal{R}}^{*}\right)}^{X}}{\u229e}}f\right(x)\star *(\neg \eta X\left(x\right)\left)\right)\left(a\right)=\sum _{x\in X}^{\mathcal{R}*}f\left(x\right)\times *\Phi (\neg \eta X(x\left)\right(a\left)\right)=\phantom{\rule{0ex}{0ex}}f\left(a\right)\times *\Phi \left({1}_{\mathcal{R}}\right)=f\left(a\right)\times *{1}_{\mathcal{R}}^{*}=f\left(a\right)$$

#### 3.2. F-Transform for $\mathcal{R}$-Fuzzy Sets

**Definition**

**9.**

**Definition**

**10.**

- 1.
- The upper F-transform of $\mathcal{R}$-fuzzy sets in X is a mapping ${F}_{\mathcal{A},X}^{\uparrow}:{\mathcal{R}}^{X}\to {\mathcal{R}}^{Y}$ defined by$$f\in {\mathcal{R}}^{X},y\in Y,\phantom{\rule{1.em}{0ex}}{F}_{\mathcal{A},X}^{\uparrow}\left(f\right)\left(y\right)=\sum _{x\in X}^{\mathcal{R}}{p}_{y}\left(x\right)\times f\left(x\right).$$
- 2.
- The lower F-transform of $\mathcal{R}$-fuzzy sets in X is a mapping ${F}_{\mathcal{A},X}^{\downarrow}:{\mathcal{R}}^{X}\to {\mathcal{R}}^{Y}$ defined by$$f\in {\mathcal{R}}^{X},y\in Y,\phantom{\rule{1.em}{0ex}}{F}_{\mathcal{A},X}^{\downarrow}\left(f\right)\left(y\right)=\sum _{x\in X}^{{\mathcal{R}}^{*}}\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}f\left(x\right).$$

**Example**

**10.**

**Theorem**

**1.**

- 1.
- The following statements are equivalent.
- (a)
- F is the $\mathcal{R}$-$po$-semimodule homomorphism and there exists a relation $u\subseteq X\times Y$ with $dom\left(u\right)=X,codom\left(u\right)=Y$ and such that$$\forall (x,y)\in u,\phantom{\rule{1.em}{0ex}}F\left({\eta}_{X}\left(x\right)\right)\left(y\right)={1}_{\mathcal{R}}.$$
- (b)
- There exists a $\mathcal{R}$-partition $\mathcal{A}=\{{p}_{y}:y\in Y\}$ such that $F={F}_{\mathcal{A},X}^{\uparrow}$.

- 2.
- The following statements are equivalent.
- (a)
- F is the ${\mathcal{R}}^{*}$-$po$-semimodule homomorphism ${\left({\mathcal{R}}^{*}\right)}^{X}\to {\left({\mathcal{R}}^{*}\right)}^{Y}$ and there exists a relation $u\subseteq X\times Y$ with $dom\left(u\right)=X,codom\left(u\right)=Y$ and such that$$\forall (x,y)\in u,\phantom{\rule{1.em}{0ex}}F(\neg \left({\eta}_{X}\left(x\right)\right)\left(y\right))={1}_{\mathcal{R}}^{*},$$
- (b)
- There exists a $\mathcal{R}$-partition $\mathcal{A}=\{{p}_{y}:y\in Y\}$ such that $F={F}_{\mathcal{A},X}^{\downarrow}$.

**Proof.**

- (1)
- Let the condition (b) holds. We set $(x,y)\in u\iff {A}_{y}\left(x\right)={1}_{\mathcal{R}}$. It is easy to that ${F}_{\mathcal{A},X}^{\uparrow}$ is the $\mathcal{R}$-$po$-semimodule homomorphism and that the additional condition holds.Let the condition (a) holds. According to Lemma 2, for arbitrary element $f\in {\mathcal{R}}^{X}$ we obtain $f={\u229e}_{x\in X}^{{\mathcal{R}}^{X}}f\left(x\right)\star {\eta}_{X}\left(x\right)$. Because F is a $\mathcal{R}$-semimodule homomorphism, according to Definition 7, we obtain$$F\left(f\right)\left(y\right)=F\left(\underset{x\in X}{\overset{{\mathcal{R}}^{X}}{\u229e}}f\right(x)\star {\eta}_{X}(x\left)\right)\left(y\right)=\sum _{x\in X}^{\mathcal{R}}f\left(x\right)\times F\left({\eta}_{X}\right(x\left)\right)\left(y\right)=\phantom{\rule{0ex}{0ex}}\sum _{x\in X}^{\mathcal{R}}f\left(x\right)\times {p}_{y}\left(x\right),$$
- (2)
- Let the condition (b) holds. For $r\in {\mathcal{R}}^{*},f,g\in {\left({\mathcal{R}}^{*}\right)}^{X}$ and $y\in Y$ we obtain$$\begin{array}{c}{F}_{\mathcal{A},X}^{\downarrow}\left(r{\star}^{*}f{+}^{*}g\right)\left(y\right)=\sum _{x\in X}^{{\mathcal{R}}^{*}}\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}\left(r{\star}^{*}f{+}^{*}g\right)\left(x\right)=\\ \sum _{x\in X}^{{\mathcal{R}}^{*}}r{\times}^{*}\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}f\left(x\right){+}^{*}\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}g\left(x\right)=\\ r{\times}^{*}{F}_{\mathcal{A},X}^{\downarrow}\left(f\right)\left(y\right){+}^{*}{F}_{\mathcal{A},X}^{\downarrow}\left(g\right)\left(y\right),\end{array}$$$$\begin{array}{c}{F}_{\mathcal{A},X}^{\downarrow}(\neg \left({\eta}_{X}\left(x\right)\right)\left(y\right)=\sum _{z\in X}^{{\mathcal{R}}^{*}}\Phi \left({p}_{y}\left(z\right)\right){\times}^{*}\Phi \left({\eta}_{X}\left(x\right)\left(z\right)\right))=\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}\Phi \left({1}_{\mathcal{R}}\right)=\\ \Phi \left({p}_{y}\left(x\right)\right){\times}^{*}{1}_{\mathcal{R}}^{*}=\Phi \left({p}_{y}\left(x\right)\right)=\Phi \left({1}_{\mathcal{R}}\right)={1}_{\mathcal{R}}^{*}.\end{array}$$Let the condition (a) holds. According to Lemma 2, we have$$\begin{array}{c}F\left(f\right)\left(y\right)=F(\underset{x\in X}{\overset{{\left({\mathcal{R}}^{*}\right)}^{X}}{\u229e}}f\left(x\right){\star}^{*}\neg \left({\eta}_{X}\left(x\right)\right))\left(y\right)=\\ \sum _{x\in X}^{{\mathcal{R}}^{*}}f\left(x\right){\times}^{*}F(\neg \left({\eta}_{X}\left(x\right)\right))\left(y\right)=\sum _{x\in X}^{{\mathcal{R}}^{*}}f\left(x\right){\times}^{*}\Phi ({p}_{y}\left(x\right))={F}_{\mathcal{A},X}^{\downarrow}(f\left)\right(y)\end{array}$$

**Definition**

**11.**

- 1.
- The upper inverse F-transform of $\mathcal{R}$-fuzzy sets defined by $\mathcal{A}$ is a mapping ${G}_{\mathcal{A},X}^{\uparrow}:{\mathcal{R}}^{Y}\to {\mathcal{R}}^{X}$, defined by$$g\in {R}^{Y},x\in X,\phantom{\rule{1.em}{0ex}}{G}_{\mathcal{A},X}^{\uparrow}\left(g\right)\left(x\right)=\sum _{y\in Y}^{{\mathcal{R}}^{*}}\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}g\left(y\right).$$
- 2.
- The lower inverse F-transform of $\mathcal{R}$-fuzzy sets defined by $\mathcal{A}$ is a mapping ${G}_{\mathcal{A},X}^{\downarrow}:{\mathcal{R}}^{Y}\to {\mathcal{R}}^{X}$, defined by$$g\in {R}^{Y},x\in X,\phantom{\rule{1.em}{0ex}}{G}_{\mathcal{A},X}^{\downarrow}\left(g\right)\left(x\right)=\sum _{y\in Y}^{\mathcal{R}}{p}_{y}\left(x\right)\times g\left(y\right).$$

**Proposition**

**2.**

- 1.
- ${F}_{\mathcal{A},X}^{\uparrow}(a\star f\u229eg)=a\star {F}_{\mathcal{A},X}^{\uparrow}\left(f\right)\u229e{F}_{\mathcal{A},X}^{\uparrow}\left(g\right)$,
- 2.
- ${F}_{\mathcal{A},X}^{\downarrow}\left(a{\star}^{*}f{\u229e}^{*}g\right)=a{\star}^{*}{F}_{\mathcal{A},X}^{\downarrow}\left(f\right){\u229e}^{*}{F}_{\mathcal{A},X}^{\downarrow}\left(g\right)$,
- 3.
- ${G}_{\mathcal{A},X}^{\uparrow}\left(a{\star}^{*}h{\u229e}^{*}k\right)=a{\star}^{*}{G}_{\mathcal{A},X}^{\uparrow}\left(h\right){\u229e}^{*}{G}_{\mathcal{A},X}^{\uparrow}\left(k\right)$,
- 4.
- ${G}_{\mathcal{A},X}^{\downarrow}(a\star h\u229ek)=a\star {G}_{\mathcal{A},X}^{\downarrow}\left(h\right)\u229e{G}_{\mathcal{A},X}^{\downarrow}\left(k\right)$,
- 5.
- ${F}_{\mathcal{A},X}^{\downarrow}\left(f\right)\left(y\right)=\Phi \left({F}_{\mathcal{A},X}^{\uparrow}(\neg f)\left(y\right)\right)$,
- 6.
- ${G}_{\mathcal{A},X}^{\downarrow}\left(g\right)\left(x\right)=\Phi \left({G}_{\mathcal{A},X}^{\uparrow}(\neg g)\left(x\right)\right)$,
- 7.
- ${G}_{\mathcal{A},X}^{\downarrow}{F}_{\mathcal{A},X}^{\downarrow}\left(f\right)\left(x\right)=\Phi \left({G}_{\mathcal{A},X}^{\uparrow}{F}_{\mathcal{A},X}^{\uparrow}(\neg f)\left(x\right)\right)$.
- 8.
- ${F}_{\mathcal{A},X}^{\uparrow}\left(f\right)\left(y\right){\ge}_{\mathcal{R}}{F}_{\mathcal{A},X}^{\downarrow}\left(f\right)\left(y\right)$
- 9.
- ${G}_{\mathcal{A},X}^{\uparrow}\left(g\right)\left(x\right){\le}_{\mathcal{R}}{G}_{\mathcal{A},X}^{\downarrow}\left(g\right)\left(x\right)$,
- 10.
- $(x,y)\in u\Rightarrow {F}_{\mathcal{A},X}^{\uparrow}\left(f\right)\left(y\right){\ge}_{\mathcal{R}}f\left(x\right)$, ${G}_{\mathcal{A},X}^{\uparrow}\left(g\right)\left(x\right){\le}_{\mathcal{R}}g\left(y\right)$,
- 11.
- $(x,y)\in u\Rightarrow {F}_{\mathcal{A},X}^{\downarrow}\left(f\right)\left(y\right){\le}_{\mathcal{R}}f\left(x\right)$, ${G}_{\mathcal{A},X}^{\downarrow}\left(g\right)\left(x\right){\ge}_{\mathcal{R}}g\left(y\right)$,

**Proof.**

**Lemma**

**3.**

**Theorem**

**2.**

- 1.
- The following statements are equivalent.
- (a)
- G is the ${\mathcal{R}}^{*}$-$po$-semimodule homomorphism ${\left({\mathcal{R}}^{*}\right)}^{Y}\to {\left({\mathcal{R}}^{*}\right)}^{X}$ and there exists a relation $u\subseteq X\times Y$ with $dom\left(u\right)=X,codom\left(u\right)=Y$ and such that$$\forall (x,y)\in u,\phantom{\rule{1.em}{0ex}}G(\neg {\eta}_{Y}\left(y\right))\left(x\right)={1}_{\mathcal{R}}^{*}.$$
- (b)
- There exists a $\mathcal{R}$-partition $\mathcal{A}=\{{p}_{y}:y\in Y\}$ such that $G={G}_{\mathcal{A},X}^{\uparrow}$.

- 2.
- The following statements are equivalent.
- (a)
- G is the $\mathcal{R}$-$po$-semimodule homomorphism ${\mathcal{R}}^{Y}\to {\mathcal{R}}^{X}$ and there exists a relation $u\subseteq X\times Y$ with $dom\left(u\right)=X,codom\left(u\right)=Y$ and such that$$\forall (x,y)\in u,\phantom{\rule{1.em}{0ex}}G\left({\eta}_{Y}\left(y\right)\right)\left(x\right)={1}_{\mathcal{R}}.$$
- (b)
- There exists a $\mathcal{R}$-partition $\mathcal{A}=\{{p}_{y}:y\in Y\}$ such that $G={G}_{\mathcal{A},X}^{\downarrow}$.

**Notation**

**2.**

- 1.
- The mappings ${F}_{X,\mathcal{A}}^{\uparrow},{F}_{X,\mathcal{A}}^{\downarrow}$ from Definition 10 are called upper and lower F-transform of this fuzzy type structure $\mathcal{F}\left(X\right)$.
- 2.
- The mapping ${G}_{Y,\mathcal{A}}^{\uparrow},{G}_{X,\mathcal{A}}^{\downarrow}$ from Definition 11 are called the upper and lower inverse F-transform of this fuzzy type structure $\mathcal{F}\left(X\right)$.

**Definition**

**12.**

**Remark**

**4.**

**Example**

**11.**

**Example**

**12.**

**Example**

**13.**

**Proposition**

**3.**

- 1.
- ${G}_{\mathcal{A},X}^{\uparrow}{F}_{\mathcal{A},X}^{\uparrow}\left(f\right)\ge f$,
- 2.
- ${G}_{\mathcal{A},X}^{\downarrow}{F}_{\mathcal{A},X}^{\downarrow}\left(f\right)\le f$,
- 3.
- ${F}_{\mathcal{A},X}^{\uparrow}{G}_{\mathcal{A},X}^{\uparrow}\left(g\right)\le g$,
- 4.
- ${F}_{\mathcal{A},X}^{\downarrow}{G}_{\mathcal{A},X}^{\downarrow}\left(g\right)\ge g$,
- 5.
- ${F}_{\mathcal{A},X}^{\uparrow}{G}_{\mathcal{A},X}^{\uparrow}{F}_{\mathcal{A},X}^{\uparrow}\left(f\right)={F}_{\mathcal{A},X}^{\uparrow}\left(f\right)$,
- 6.
- ${G}_{\mathcal{A},X}^{\downarrow}{F}_{\mathcal{A},X}^{\downarrow}{G}_{\mathcal{A},X}^{\downarrow}\left(g\right)={G}_{\mathcal{A},X}^{\downarrow}\left(g\right)$.
- 7.
- ${F}_{\mathcal{A},X}^{\uparrow}\left(f\right){\le}_{Y}g\Rightarrow f{\le}_{X}{G}_{\mathcal{A},X}^{\uparrow}\left(g\right)$,
- 8.
- ${G}_{\mathcal{A},X}^{\downarrow}\left(g\right){\le}_{X}f\Rightarrow g{\le}_{Y}{F}_{\mathcal{A},X}^{\downarrow}\left(f\right)$.

**Proof.**

- (1)
- According to axiom (+) we have$$\begin{array}{c}{G}^{\uparrow}{F}^{\uparrow}\left(f\right)\left(x\right)=\sum _{y\in Y}^{{\mathcal{R}}^{*}}(\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}{F}^{\uparrow}\left(f\right)\left(y\right))=\end{array}$$$$\begin{array}{c}\sum _{y\in Y}^{{\mathcal{R}}^{*}}(\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}(\sum _{z\in X}^{\mathcal{R}}{p}_{y}\left(z\right)\times f\left(z\right)))=\end{array}$$$$\begin{array}{c}\sum _{y\in Y}^{{\mathcal{R}}^{*}}\sum _{z\in X}^{\mathcal{R}}\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}({p}_{y}\left(z\right)\times f\left(z\right)).\end{array}$$Because $\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}({p}_{y}\left(z\right)\times f\left(z\right)\in R$, according to Definition 2(v) we have $\Phi \left({p}_{y}\left(x\right)\right)$${\times}^{*}({p}_{y}\left(z\right)\times f\left(z\right)\ge 0$ and, according to axiom (+), for arbitrary $y\in Y$ it follows$$\begin{array}{c}\sum _{z\in X}^{\mathcal{R}}(\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}({p}_{y}\left(z\right)\times f\left(z\right)))\ge \Phi \left({p}_{y}\left(x\right)\right){\times}^{*}({p}_{y}\left(x\right)\times f\left(x\right))\ge f\left(x\right).\end{array}$$According to Definition 5, for arbitrary $y\in Y$ we have$$\sum _{z\in X}^{\mathcal{R}}(\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}({p}_{y}\left(z\right)\times f\left(z\right))){\le}^{*}\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}({p}_{y}\left(x\right)\times f\left(x\right)){\le}^{*}f\left(x\right).$$Therefore, using inequalities (3), (4), (5) and (6), we have$$\begin{array}{c}{G}^{\uparrow}{F}^{\uparrow}\left(f\right)\left(x\right){\le}^{*}\sum _{y\in Y}^{{\mathcal{R}}^{*}}\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}({p}_{y}\left(x\right)\times f\left(x\right)){\le}^{*}\\ \sum _{y\in Y}^{{\mathcal{R}}^{*}}f\left(x\right)=f\left(x\right){\times}^{*}\left(\sum _{y\in Y}{1}^{*}\right)=f\left(x\right){\times}^{*}{1}^{*}=f\left(x\right).\end{array}$$
- (2)
- According to Proposition 2 and previous part (1), we have$$\begin{array}{c}{G}^{\downarrow}{F}^{\downarrow}\left(f\right)\left(x\right)=\Phi \left({G}^{\uparrow}{F}^{\uparrow}(\neg f)\right)\left(x\right){\ge}^{*}\Phi (\Phi \left(f\left(x\right)\right))=f\left(x\right),\end{array}$$
- (3)
- From the axiom (+) and its dual version, analogously as in (1), it follows$$\begin{array}{c}{F}^{\uparrow}{G}^{\uparrow}\left(g\right)\left(y\right)=\sum _{x\in X}^{\mathcal{R}}{p}_{y}\left(x\right)\times {G}^{\uparrow}\left(g\right)\left(x\right)=\\ \sum _{x\in X}^{\mathcal{R}}\sum _{t\in Y}^{{\mathcal{R}}^{*}}{p}_{y}\left(x\right)\times (\Phi \left({p}_{t}\left(x\right)\right){\times}^{*}g\left(t\right))\le \\ \sum _{x\in X}^{\mathcal{R}}{p}_{y}\left(x\right)\times (\Phi \left({p}_{y}\left(x\right)\right){\times}^{*}g\left(y\right))\le \sum _{x\in X}^{\mathcal{R}}g\left(y\right)=g\left(y\right).\end{array}$$
- (4)
- The proof can be analogously as in (2) and it will be omitted.
- (5)
- From (3) it follows ${F}^{\uparrow}{G}^{\uparrow}{F}^{\uparrow}\left(f\right)\le {F}^{\uparrow}\left(f\right)$ and because ${F}^{\uparrow}$ is order preserving, from ${G}^{\uparrow}{F}^{\uparrow}\left(f\right)\ge f$ we obtain the other inequality.
- (6)
- From the property (3) it follows ${G}^{\downarrow}{F}^{\downarrow}\left({G}^{\downarrow}\left(g\right)\right)\le {G}^{\downarrow}\left(g\right)$. Because ${G}^{\downarrow}$ preserves the ordering ≤, from the property (4) it follows ${G}^{\downarrow}\left({F}^{\downarrow}{G}^{\downarrow}\left(g\right)\right)\ge {G}^{\downarrow}\left(g\right)$.
- (7)
- From ${F}^{\uparrow}\left(f\right){\le}_{Y}g$ it follows ${F}^{\uparrow}\left(f\right){\ge}_{Y}^{*}g$ and because ${G}^{\uparrow}$ preserves ${\le}_{Y}^{*}$-ordering, from (3) we obtain $f{\ge}_{X}^{*}{G}^{\uparrow}{F}^{\uparrow}\left(f\right){\ge}_{X}^{*}{G}^{\uparrow}\left(g\right)$. Hence, ${G}^{\uparrow}\left(g\right){\ge}_{X}f$.
- (8)
- The proof can be done analogously as in (7) and it will be omitted.

**Corollary**

**1.**

- 1.
- For arbitrary fuzzy type structure which is transformable to $\mathcal{R}$-fuzzy sets, the F-transform and inverse F-transform of this fuzzy type structure satisfy all properties from Proposition 2.
- 2.
- The F-transform and inverse F-transform of $\mathcal{L}$-fuzzy soft sets satisfy all properties of Proposition 3.

## 4. Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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Močkoř, J.
Semiring-Valued Fuzzy Sets and F-Transform. *Mathematics* **2021**, *9*, 3107.
https://doi.org/10.3390/math9233107

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Močkoř J.
Semiring-Valued Fuzzy Sets and F-Transform. *Mathematics*. 2021; 9(23):3107.
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2021. "Semiring-Valued Fuzzy Sets and F-Transform" *Mathematics* 9, no. 23: 3107.
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