# M-Hazy Vector Spaces over M-Hazy Field

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**Assume that $\diamond :M\times M\u27f6M$ is a function. ⋄ is defined to be a triangular norm (for short, t-norm) on $M,$ if the following conditions holds:

- (1)
- $u\diamond v=v\diamond u,$
- (2)
- $(u\diamond v)\diamond w=u\diamond (v\diamond w),$
- (3)
- $u\le w,$$v\le x$ implies $u\diamond v\le w\diamond x,$
- (4)
- $u\diamond 1=u$ for all $u\in M.$

**Definition**

**2**

**.**Assume that $\to :M\times M\u27f6M$ is a function, and ⋄ is a t-norm in $M.$ Then, → is defined to be the residuum of ⋄, if, for all $u,$$v,$$w\in M,$

**Definition**

**3**

**.**Assume that $(M,\vee ,\wedge ,\perp ,\top )$ is a bounded lattice, where ⊥ represents the least element, ⊤ the greatest element, ⋄ is a t-norm on M, and → denotes the residuum of ⋄. Then, $(M,\vee ,\wedge ,\diamond ,\to ,\perp ,\top )$ is said to be a residuated lattice.

**Proposition**

**1**

**.**Assume $(M,\vee ,\wedge ,\diamond ,\to ,\perp ,\top )$ is a completely residuated lattice. Then, for every $u,$$v,$$w\in M,$${\left\{{u}_{i}\right\}}_{i\in I},$${\left\{{v}_{i}\right\}}_{i\in I}\subseteq M$, the below statements are valid:

- (1)
- $u\to v=\bigvee \{w\in M\mid u\diamond w\le v\}.$
- (2)
- $v\le u\to v,$$\top \to u=u.$
- (3)
- $u\to ({\displaystyle \underset{i\in I}{\bigwedge}}{v}_{i})={\displaystyle \underset{i\in I}{\bigwedge}}(u\to {v}_{i}).$
- (4)
- $({\displaystyle \underset{i\in I}{\bigvee}}{u}_{i})\to v={\displaystyle \underset{i\in I}{\bigwedge}}({u}_{i}\to v).$

**Definition**

**4**

**.**Assume that $\ast :P\times P\u27f6{M}^{P}$ is a function; then, * is defined to be an M-hazy operation on P, if the conditions given below hold:

**(MH1)**- $\forall u,$$v\in P,$ we have $\underset{p\in P}{\bigvee}}(u\ast v)\left(p\right)\ne \perp .$
**(MH2)**- $\forall u,$$v,$$p,$$q\in P,$$(u\ast v)\left(p\right)\diamond (u\ast v)\left(q\right)\ne \perp \Rightarrow p=q.$

**Definition**

**5**

**.**Assume $\ast :P\times P\u27f6{M}^{P}$ is an M-hazy operation on a nonempty set $P.$ Then, $(P,\ast )$ is defined to be an M-hazy group (in short, MHG) if the following conditions hold:

**(MG1)**- $\forall u,$$v,$$w,$$p,$$q\in P,$$(u\ast v)\left(p\right)\diamond (v\ast w)\left(q\right)\le {\displaystyle \underset{r\in P}{\bigwedge}}((p\ast w)\left(r\right)\leftrightarrow (u\ast q)\left(r\right)),$ i.e., the M-hazy associative law holds.
**(MG2)**- An element $o\in P$ is said to be the left identity element of P, if $o\ast u={u}_{\top}$ for all $u\in P.$
**(MG3)**- An element $v\in P$ is said to be the left inverse of u, if for each $u\in P$, $v\ast u={o}_{\top},$ and is denoted by ${u}^{-1}.$$(P,\ast )$ is defined to be an abelian MHG if the following condition holds:
**(MG4)**- $u\ast v=v\ast u$ for all $u,$$v\in P.$

**Definition**

**6**

**.**Assume $+:R\times R\u27f6{M}^{R}$ and $\u2022:R\times R\u27f6{M}^{R}$ are the M-hazy addition operation and M-hazy multiplication operation on R, respectively. Then, $(R,+,\u2022)$ is defined to be an M-hazy ring (in short, MHR) if the below conditions hold:

**(MHR1)**- $(R,+)$ is an abelian MHG.
**(MHR2)**- $(R,\u2022)$ is an M-hazy semigroup.
**(MHR3)**- $\forall u,$$v,$$w,$$p,$$q,$$r\in R,(u\u2022v)\left(p\right)\diamond (v+w)\left(q\right)\diamond (u\u2022w)\left(r\right)\le {\displaystyle \underset{s\in R}{\bigwedge}}((u\u2022q)\left(s\right)\leftrightarrow (p+r)\left(s\right)).$

**Definition**

**7**

**.**A function $\mathcal{S}:{2}^{P}\to M$ is said to be an M-fuzzifying convexity on a nonempty set P if the below conditions hold:

- (1)
- $\mathcal{S}(\varnothing )=\mathcal{S}\left(P\right)=\top $;
- (2)
- If $\left\{{D}_{i}\right|i\in \Omega \}\subseteq {2}^{P}$ is nonempty, then $\mathcal{S}\left({\bigcap}_{i\in \Omega}{D}_{i}\right)\ge {\bigwedge}_{i\in \Omega}\mathcal{S}\left({D}_{i}\right)$;
- (3)
- If $\left\{{D}_{i}\right|i\in \Omega \}\stackrel{dir}{\subseteq}{2}^{P}$, then $\mathcal{S}\left({\bigcup}_{i\in \Omega}{D}_{i}\right)\ge {\bigwedge}_{i\in \Omega}\mathcal{S}\left({D}_{i}\right)$.

**Definition**

**8**

**.**A field is a set F with two operations, called addition and multiplication, which satisfy the following conditions:

**(F1)**- $\forall u,v\in F,u+v=v+u$,
**(F2)**- $\forall u,v,w\in F,(u+v)+w=u+(v+w)$,
**(F3)**- F contains an element 0 such that $0+u=u,\forall u\in F$,
**(F4)**- For each $u\in F$, there is an element $-u\in F$ such that $u+(-u)=0$,
**(F5)**- $\forall u,v\in F\Rightarrow u\xb7v\in F$,
**(F6)**- $\forall u,v\in F,u\xb7v=v\xb7u$,
**(F7)**- $\forall u,v,w\in F,(u\xb7v)\xb7w=u\xb7(v\xb7w)$,
**(F8)**- F contains an element $1\ne 0$ and $\forall u\in F$ such that $1\xb7u=u$,
**(F9)**- For each $0\ne u\in F$, there is an element ${u}^{-1}\in F$ such that $u\xb7{u}^{-1}=1$,
**(F10)**- $\forall u,v,w\in F,u\xb7(v+w)=u\xb7w+v\xb7w$.

**Definition**

**9**

**.**A vector space is a nonempty set V over a field F, whose objects are called vectors equipped with two operations, called addition and scalar multiplication: for any two vectors $u,v$ in V and a scalar a in F defined by the mappings $+:V\times V\u27f6V$ and $\xb7:F\times V\u27f6V$, the following conditions are satisfied:

**(V1)**- $\forall u,v,w\in V,(u+v)+w=u+(v+w)$,
**(V2)**- There is a vector 0, called the zero vector, such that $u+0=u$,
**(V3)**- For any vector u, there is a vector $-u$ such that $u+(-u)=0$,
**(V4)**- $\forall u,v\in V,u+v=v+u$,
**(V5)**- $\forall u,v\in V,\forall a\in F,a\xb7(u+v)=a\xb7u+a\xb7v$,
**(V6)**- $\forall a,b\in F,\forall u\in V,(a+b)\xb7u=a\xb7u+b\xb7u$,
**(V7)**- $\forall a,b\in F,\forall u\in V,a\xb7(b\xb7u)=(a\xb7b)\xb7u$,
**(V8)**- $\forall 1\in F,\forall u\in V,1\xb7u=u.$

## 3. M-Hazy Vector Spaces

**Definition**

**10.**

**(MHF1)**- $(F,+)$ is an abelian MHG.
**(MHF2)**- $(F,\u2022)$ is an abelian MHG.
**(MHF3)**- $\forall u,$$v,$$w,$$p,$$q,$$r\in F,(u\u2022v)\left(p\right)\diamond (v+w)\left(q\right)\diamond (u\u2022w)\left(r\right)\le {\displaystyle \underset{s\in F}{\bigwedge}}((u\u2022q)\left(s\right)\leftrightarrow (p+r)\left(s\right)).$

**Proposition**

**2.**

- (1)
- $o+u=u+o={u}_{\top}$.
- (2)
- $e\u2022u=u\u2022e={u}_{\top}$.

**Proof.**

**Proposition**

**3.**

- (1)
- $(-u)+u=u+(-u)={o}_{\top}.$
- (2)
- ${u}^{-1}\u2022u=u\u2022{u}^{-1}={e}_{\top}$.

**Proof.**

**Example**

**1.**

**Proposition**

**4.**

- (1)
- $\begin{array}{cc}& (u+v)\left(w\right)=\left(\right(-u)+w)\left(v\right)=(w+(-v\left)\right)\left(u\right)=(v+(-w\left)\right)(-u)\hfill \\ & =\left(\right(-w)+u)(-v)=\left(\right(-v)+(-u\left)\right)(-w).\hfill \end{array}$
- (2)
- $\begin{array}{cc}& (u\u2022v)\left(w\right)=({u}^{-1}\u2022w)\left(v\right)=(w\u2022{v}^{-1})\left(u\right)=(v\u2022{w}^{-1})\left({u}^{-1}\right)=({w}^{-1}\u2022u)\left({v}^{-1}\right)\hfill \\ & =({v}^{-1}\u2022{u}^{-1})\left({w}^{-1}\right).\hfill \end{array}$

**Proof.**

**Definition**

**11.**

**(MHV1)**- $\forall u,v,$$p,$$q,$$r\in V$ and $a\in F,(a\circ u)\left(p\right)\diamond (u\oplus v)\left(q\right)\diamond (a\circ v)\left(r\right)\le {\displaystyle \underset{s\in V}{\bigwedge}}((a\circ q)\left(s\right)\leftrightarrow (p\oplus r)\left(s\right)).$
**(MHV2)**- $\forall u,p,q\in V$ and $a,b\in F,(a\circ u)\left(p\right)\diamond (a+b)\left(q\right)\diamond (b\circ u)\left(r\right)\le {\displaystyle \underset{s\in V}{\bigwedge}}((q\circ u)\left(s\right)\leftrightarrow (p\oplus r)\left(s\right)).$
**(MHV3)**- $\forall u,$$q\in V,$ and $\forall a,b,c\in F$, $(a\u2022b)\left(c\right)\diamond (b\circ u)\left(q\right)\le {\displaystyle \underset{r\in V}{\bigwedge}}((c\circ u)\left(r\right)\leftrightarrow (a\circ q)\left(r\right)).$
**(MHV4)**- $\forall u\in V$ and $e\in F$, $e\circ u={e}_{\top}.$

**Proposition****5.****(MHV1)**- $\forall u,v,p,q,r\in V$, and $\forall a\in F$,$$\begin{array}{c}(a\circ u)\left(p\right)\diamond (u\oplus v)\left(q\right)\diamond (a\circ v)\left(r\right)\le {\displaystyle \underset{s\in V}{\bigwedge}}((a\circ q)\left(s\right)\leftrightarrow (p\oplus r)\left(s\right)).\hfill \end{array}$$
**(MHV1${}^{\prime}$)**- $\forall u,v,p,q,r,s\in V$, and $\forall a\in F$,$$\begin{array}{c}(a\circ u)\left(p\right)\diamond (u\oplus v)\left(q\right)\diamond (a\circ v)\left(r\right)\diamond (a\circ q)\left(s\right)\le (p\oplus r)\left(s\right)\hfill \end{array}$$$$\begin{array}{c}(a\circ u)\left(p\right)\diamond (u\oplus v)\left(q\right)\diamond (a\circ v)\left(r\right)\diamond (p\oplus r)\left(s\right)\le (a\circ q)\left(s\right).\hfill \end{array}$$
**(MHV1${}^{\u2033}$)**- $$\begin{array}{c}Ifa\circ u={p}_{\lambda},u\oplus v={q}_{\mu},a\circ v={r}_{\nu},\hfill \end{array}$$
**(MHV1${}^{\u2034}$)**- If $a\circ u={p}_{\lambda}$, $u\oplus v={q}_{\mu}$, $a\circ v={r}_{\nu}$, $a\circ q={t}_{\alpha}$, $p\oplus r={u}_{\beta}$, then$$t=u,\lambda \diamond \mu \diamond \nu \diamond \alpha \le \beta \phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\lambda \diamond \mu \diamond \nu \diamond \beta \le \alpha .$$

**Proof.**

**Example**

**2.**

- (1)
- The Euclidean space ${R}^{n}$ is an M-hazy vector space under the addition and scalar multiplication.
- (2)
- The set ${P}_{n}$ of all polynomials of degree less than or equal to n is an M-hazy vector space under the addition and scalar multiplication of polynomials.
- (3)
- The set $M(m,n)$ of all $m\times n$ matrices is an M-hazy vector space under the addition and scalar multiplication of matrices.

**Theorem**

**1.**

- (1)
- $(a\circ o)\left(o\right)\ne \perp .$
- (2)
- $(o\circ u)\left(o\right)\ne \perp .$
- (3)
- If $(a\circ u)\left(o\right)\ne \perp ,$ then $a=o$ or $u=o$.

**Proof.**

## 4. M-Hazy Subspaces

**Definition**

**12.**

**Theorem**

**2.**

- (1)
- $\forall u,v\in W$, we have $\underset{p\in W}{\bigvee}}(u\oplus v)\left(p\right)\ne \perp $,
- (2)
- $\forall u\in W$ and $\forall a\in F$, we have $\underset{p\in W}{\bigvee}}(a\circ u)\left(p\right)\ne \perp $,
- (3)
- $\forall u\in W$, we have $-u\in W$.

**Proof.**

**Theorem**

**3.**

- (1)
- $\forall u,v\in W$, we have ${\displaystyle \underset{p\in W}{\bigvee}}}(u\oplus (-v))\left(p\right)\ne \perp $,
- (2)
- $\forall u\in W$ and $\forall a\in F$, we have ${\displaystyle \underset{p\in W}{\bigvee}}}(a\circ u)\left(p\right)\ne \perp $.

**Proof.**

**Theorem**

**4.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

## 5. Linear Transformation of M-Hazy Vector Spaces

**Definition**

**13.**

- (1)
- $\forall u,v\in V,{\tau}_{M}^{\to}(u\oplus v)=(\tau \left(u\right)\u229e\tau \left(v\right)),$
- (2)
- $\forall u\in V,\forall a\in F,{\tau}_{M}^{\to}(a\circ u)=(a\odot \tau \left(u\right)).$

**Definition**

**14.**

**Example**

**3.**

- (1)
- Assume that $(V,\oplus ,\circ ,F)$ is an M-hazy vector space over an M-hazy field $(F,+,\u2022)$, the set $\left\{o\right\}$ and the whole M-hazy vector space V are M-hazy subspaces of V; they are called the trivial M-hazy subspaces of V.
- (2)
- Assume ${R}^{n}$ and ${R}^{m}$ are the Euclidean spaces and $\tau :{R}^{n}\u27f6{R}^{m}$ is a linear transformation. The image$${\tau}^{\to}\left(p\right)=\{\tau \left(p\right):p\in {R}^{n}\}$$$${\tau}^{\leftarrow}\left(\left\{o{}^{\prime}\right\}\right)=\{p\in {R}^{n}\mid \tau \left(p\right)=o{}^{\prime}\}$$

**Proposition**

**9.**

- (1)
- If J is an M-hazy subspace of V, then ${\tau}^{\to}\left(J\right)$ is an M-hazy subspace of W.
- (2)
- If K is an M-hazy subspace of W, then ${\tau}^{\leftarrow}\left(K\right)$ is an M-hazy subspace of V containing $ker\tau $.

**Proof.**

**Proposition**

**10.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

- (1)
- $\forall u,v\in V,{\tau}_{M}^{\to}(u\oplus v)=(\tau \left(u\right)\u229e\tau \left(v\right)),$
- (2)
- $\forall u\in V,\forall a\in F,{\tau}_{M}^{\to}(a\circ u)=(a\odot \tau \left(u\right)).$

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Mehmood, F.; Shi, F.-G.
*M*-Hazy Vector Spaces over *M*-Hazy Field. *Mathematics* **2021**, *9*, 1118.
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Mehmood, Faisal, and Fu-Gui Shi.
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