**2. Preliminaries**

This section contains the fundamental definitions about completely residuated lattices, fields, vector spaces, vector subspaces, *M*-hazy groups, *M*-hazy rings, and *M*fuzzifying convexity.

All through this paper, (*<sup>M</sup>*, ∨, ∧, , →, ⊥, ) represents a completely residuated lattice, and ≤ denotes the partial order of *M*. Assume *P* is a nonempty set and *K* ⊆ *M*, then *K* denotes the least upper bound of *K* and *K* the greatest lower bound of *K*. 2*<sup>P</sup>* (resp., *M<sup>P</sup>*) denotes the collection of all subsets (resp., *M*-subsets) on *P*. A family {*Ai* | *i* ∈ *ω*} is up-directed provided for each *A*1, *A*2 ∈ {*Ai* | *i* ∈ *ω*} that there exists a third element *dir*

*A*3 ∈ {*Ai* | *i* ∈ *ω*} such that *A*1 ⊆ *A*3 and *A*2 ⊆ *A*3 are denoted by: {*Ai* | *i* ∈ *ω*} ⊆ 2*P*.

**Definition 1** ([55])**.** *Assume that* : *M* × *M* −→ *M is a function. is defined to be a triangular norm (for short, t-norm) on M*, *if the following conditions holds:*


**Definition 2** ([55])**.** *Assume that* →: *M* × *M* −→ *M is a function, and is a t-norm in M*. *Then,* → *is defined to be the residuum of , if, for all u*, *v*, *w* ∈ *M*,

$$u \le v \to w \Leftrightarrow u \diamond v \le w.$$

**Definition 3** ([56])**.** *Assume that* (*<sup>M</sup>*, ∨, ∧, ⊥, ) *is a bounded lattice, where* ⊥ *represents the least element, the greatest element, is a t-norm on M, and* → *denotes the residuum of . Then,* (*<sup>M</sup>*, ∨, ∧, , →, ⊥, ) *is said to be a residuated lattice.*

A residuated lattice is defined to be a completely residuated lattice if the primary lattice is complete. In addition, we define *u* ↔ *v* = (*u* → *v*) ∧ (*v* → *<sup>u</sup>*). The proposition below shows properties of the implication operation.

**Proposition 1** ([57,58])**.** *Assume* (*<sup>M</sup>*, ∨, ∧, , →, ⊥, ) *is a completely residuated lattice. Then, for every u*, *v*, *w* ∈ *M*, {*ui*}*<sup>i</sup>*∈*I*, {*vi*}*<sup>i</sup>*∈*<sup>I</sup>* ⊆ *M, the below statements are valid:*



**Definition 4** ([11])**.** *Assume that* ∗ : *P* × *P* −→ *M<sup>P</sup> is a function; then,* ∗ *is defined to be an M-hazy operation on P, if the conditions given below hold:*

**(MH1)** ∀*<sup>u</sup>*, *v* ∈ *P*, *we have p*∈*P* (*u* ∗ *v*)(*p*) = ⊥. **(MH2)**∀*<sup>u</sup>*,*v*,*p*,*q*∈*P*,(*u*∗ *v*)(*p*)(*u*∗*v*)(*q*)=⊥⇒*p*

**Definition 5** ([11])**.** *Assume* ∗ : *P* × *P* −→ *M<sup>P</sup> is an M-hazy operation on a nonempty set P*. *Then,* (*<sup>P</sup>*, ∗) *is defined to be an M-hazy group (in short, MHG) if the following conditions hold:*

 = *q*.


(*<sup>P</sup>*, ∗) *is defined to be an abelian MHG if the following condition holds:*

**(MG4)** *u* ∗ *v* = *v* ∗ *u for all u*, *v* ∈ *P*.

**Definition 6** ([12])**.** *Assume* + : *R* × *R* −→ *M<sup>R</sup> and* • : *R* × *R* −→ *M<sup>R</sup> are the M-hazy addition operation and M-hazy multiplication operation on R, respectively. Then,* (*<sup>R</sup>*, +, •) *is defined to be an M-hazy ring (in short, MHR) if the below conditions hold:*


**(MHR3)** ∀*<sup>u</sup>*, *v*, *w*, *p*, *q*, *r* ∈ *<sup>R</sup>*,(*<sup>u</sup>* • *v*)(*p*) (*v* + *w*)(*q*) (*u* • *w*)(*r*) ≤ *s*∈*R* ((*u* • *q*)(*s*) ↔ (*p* + *<sup>r</sup>*)(*s*)).

We now give the definition of *M*-fuzzifying convex space, and we refer to Vel [32] for all of the background on the convexity theory that may be required.

**Definition 7** ([34])**.** *A function* S : 2*<sup>P</sup>* → *M is said to be an M-fuzzifying convexity on a nonempty set P if the below conditions hold:*


A function *π* : (*<sup>P</sup>*, S*P*) −→ (*Q*, <sup>S</sup>*Q*) is defined as *M*-fuzzifying convexity preserving (*M*-CP, in short) given that S*P*(*π*<sup>←</sup>(*B*)) ≥ <sup>S</sup>*Q*(*B*) for each *B* ∈ <sup>2</sup>*Q*; *π* is called *M*-fuzzifying convex-to-convex (*M*-CC, in short) provided that <sup>S</sup>*Q*(*π*<sup>→</sup>(*A*)) ≥ S*P*(*A*) for each *A* ∈ 2*P*.

**Definition 8** ([22])**.** *A field is a set F with two operations, called addition and multiplication, which satisfy the following conditions:*


**Definition 9** ([22])**.** *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* × *V* −→ *V and* · : *F* × *V* −→ *V, the following conditions are satisfied:*


#### **3.** *M***-Hazy Vector Spaces**

In this section, we introduce the concept of *M*-hazy vector spaces over the *M*-hazy field. We first introduce the concept of *M*-hazy field and give its properties, which are necessary to present the concept of *M*-hazy vector space.

**Definition 10.** *Assume* + : *F* × *F* −→ *M<sup>F</sup> and* • : *F* × *F* −→ *M<sup>F</sup> are the M-hazy addition operation and M-hazy multiplication operation on F, respectively. Then, the 3-tuple* (*<sup>F</sup>*, +, •) *is defined to be an M-hazy field (in short, MHF) if the below conditions hold:*

**(MHF1)** (*<sup>F</sup>*, +) *is an abelian MHG.*

**(MHF2)** (*<sup>F</sup>*, •) *is an abelian MHG.*

**(MHF3)** ∀*<sup>u</sup>*, *v*, *w*, *p*, *q*, *r* ∈ *<sup>F</sup>*,(*<sup>u</sup>* • *v*)(*p*) (*v* + *w*)(*q*) (*u* • *w*)(*r*) ≤ *s*∈*F* ((*u* • *q*)(*s*) ↔ (*p* + *<sup>r</sup>*)(*s*)).

**Proposition 2.** *Assume that* (*<sup>F</sup>*, +, •) *is an M-hazy field, and o and e are the additive and multiplicative identity elements of F, respectively. Then,* ∀*u* ∈ *F;*

*(1) o* + *u* = *u* + *o* = *u. (2)e*•*u*= *u* •*e*=*u.*

**Proof.** The proof is similar to the proof of Proposition 3.8 in [11] so it is omitted.

**Proposition 3.** *In an M-hazy field* (*<sup>F</sup>*, +, •)*, the left additive inverse* −*u of u is also the right additive inverse of u in* (*<sup>F</sup>*, +, •)*. In addition, the left multiplicative inverse u*<sup>−</sup><sup>1</sup> *of u is also the right multiplicative inverse of u in* (*<sup>F</sup>*, +, •)*. That is, the following conditions hold:*

*(1)* (−*<sup>u</sup>*) + *u* = *u* + (−*<sup>u</sup>*) = *o*.*(2) u*<sup>−</sup><sup>1</sup> • *u* = *u* • *u*<sup>−</sup><sup>1</sup> = *e.*

**Proof.** The proof is similar to the proof of Proposition 3.7 in [11] so it is omitted.

**Example 1.** *Assume that F* = {*<sup>o</sup>*,*e*, *u*, *v*} *is a set and assume* (*<sup>M</sup>*, ) = ([0, 1], <sup>∧</sup>)*. The mappings* +: *F* × ×*F* −→ [0, 1]*<sup>F</sup> and* • : *F* × *F* −→ [0, 1]*<sup>F</sup> are defined by the following tables:*


(b) Values of the [0, 1]-hazy operation •.


*It is easy to verify (MHF1), (MHF2), and (MHF3) analogous to Example 3.3 in [12].* **Proposition 4.** *Assume that* (*<sup>F</sup>*, +, •) *is an M-hazy field; then, the following equations hold:*

$$\begin{array}{l} (1) \qquad \quad (u+v)(w) = ((-u)+w)(v) = (w+(-v))(u) = (v+(-w))(-u) \\ \qquad \quad = ((-w)+u)(-v) = ((-v)+(-u))(-w). \\ (u\bullet v)(w) = (u^{-1}\bullet w)(v) = (w\bullet v^{-1})(u) = (v\bullet w^{-1})(u^{-1}) = (w^{-1}\bullet u)(v^{-1}) \\ \qquad \quad = (v^{-1}\bullet u^{-1})(w^{-1}). \end{array}$$

**Proof.** The proof is similar to the proof of Proposition 3.11 and Corollary 3.12 in [11] so it is omitted.

We now present the concept of *M*-hazy vector space.

**Definition 11.** *Assume* (*<sup>F</sup>*, +, •) *is an M-hazy field and* (*<sup>V</sup>*, ⊕) *is an abelian M-hazy group. We define an M-hazy vector space over F as a quadruple* (*<sup>V</sup>*, ⊕, ◦, *<sup>F</sup>*)*, where* ◦ *is a mapping* ◦ : *F* × *V* −→ *M<sup>V</sup> such that the following conditions hold:*

**(MHV1)** ∀*<sup>u</sup>*, *v, p*, *q*, *r* ∈ *V and a* ∈ *<sup>F</sup>*,(*<sup>a</sup>* ◦ *u*)(*p*) (*u* ⊕ *v*)(*q*) (*a* ◦ *v*)(*r*) ≤ *s*∈*V* ((*a* ◦ *q*)(*s*) ↔ (*p*⊕*<sup>r</sup>*)(*s*)).

**(MHV2)** ∀*<sup>u</sup>*, *p*, *q* ∈ *V and a*, *b* ∈ *<sup>F</sup>*,(*<sup>a</sup>* ◦ *u*)(*p*) (*a* + *b*)(*q*) (*b* ◦ *u*)(*r*) ≤ *s*∈*V* ((*q* ◦ *u*)(*s*) ↔ (*p<sup>r</sup>*)(*s*)).

$$(\mathbf{M} \mathbf{H} \mathbf{V} \mathbf{3}) \; \forall u, q \in V, and \; \forall a, b, c \in F, \\
(a \bullet b)(c) \diamond (b \circ u)(q) \le \bigwedge\_{r \in V} ((c \circ u)(r) \leftrightarrow (a \circ q)(r)).$$

**(MHV4)** ∀*u* ∈ *V and e* ∈ *F, e* ◦ *u* = *e*.

=

 ⊕

**Proposition 5.** *Assume* ⊕ : *V* × *V* −→ *M<sup>V</sup> and* ◦ : *F* × *V* −→ *M<sup>V</sup> are the M-hazy operations under addition and under scalar multiplication on V, respectively; then, the following statements are equivalent for all u*, *v* ∈ *V, and* ∀*a* ∈ *F.*

**(MHV1)** ∀*<sup>u</sup>*, *v*, *p*, *q*,*<sup>r</sup>* ∈ *V, and* ∀*a* ∈ *F,*

$$((a \circ u)(p) \diamond (u \oplus v)(q) \diamond (a \circ v)(r) \le \bigwedge\_{s \in V} ((a \circ q)(s) \leftrightarrow (p \oplus r)(s)).$$

**(MHV1)** ∀*<sup>u</sup>*, *v*, *p*, *q*,*r*,*<sup>s</sup>* ∈ *V, and* ∀*a* ∈ *F,*

$$(a \circ u)(p) \diamond (u \oplus v)(q) \diamond (a \diamond v)(r) \diamond (a \diamond q)(s) \le (p \oplus r)(s)$$

*and*

$$(a \circ u)(p) \diamond (u \oplus v)(q) \diamond (a \circ v)(r) \diamond (p \oplus r)(s) \le (a \circ q)(s).$$

**(MHV1)**

$$If a \circ \iota = p\_{\lambda \nu} \iota \oplus \upsilon = q\_{\mu \nu} a \diamond \upsilon = r\_{\nu \nu}$$

*then* (*a* ◦ *q*) *λ μ ν* ≤ (*p* ⊕ *r*) *and* (*p* ⊕ *r*) *λ μ ν* ≤ (*a* ◦ *q*). **(MHV1)***Ifa*◦*u*=*pλ,u*⊕*v*=*qμ,a* ◦ *v* =*rν,a*◦*q*=*t<sup>α</sup>,p*⊕*r*=*<sup>u</sup>β,then*

 *t* = *u*, *λ μ ν α* ≤ *β and λ μ ν β* ≤ *α*.

**Proof.** (MHV1) ⇒ (MHV1) The proof is simple so it is omitted. (MHV1) ⇒ (MHV1) ∀*a* ∈ *F* and ∀*q* ∈ *V*, we have

> ((*a* ◦ *q*) *λ μ ν*)(*s*) = (*a* ◦ *q*)(*s*) *λ μ ν* = (*a* ◦ *q*)(*s*) (*a* ◦ *u*)(*p*) (*u* ⊕ *v*)(*q*) (*a* ◦ *v*)(*r*) ≤ (*p* ⊕ *<sup>r</sup>*)(*s*),

that is, (*a* ◦ *q*) *λ μ ν* ≤ *p* ⊕ *r*. A similar argumen<sup>t</sup> shows the other inequality.

(MHV1) ⇒ (MHV1) We need to only verify *t* = *u*. According to (MHV1), we have *tα λ μ ν* ≤ *<sup>u</sup>β*, that is, *<sup>t</sup>αλμν* ≤ *<sup>u</sup>β*, whence by (MH2), *t* = *u*. (MHV1) ⇒ (MHV1) ∀*p*, *q*,*r*,*<sup>s</sup>* ∈ *P* and ∀*a* ∈ *F*, we have

> (*a* ◦ *u*)(*p*) (*u* ⊕ *v*)(*q*) (*a* ◦ *v*)(*r*) = *λ μ ν* ≤ (*α* → *β*) ∧ (*β* → *α*) = *α* ↔ *β* = (*a* ◦ *q*)(*t*) ↔ (*p* ⊕ *r*)(*u*)

and, by (MH2), we can complete the proof.


In the following discussion, we assume that the operation in the completely residuated lattice *M* is ∧; that is, the lattice valued environment *M* is degenerated to complete Heyting algebra. We also assume that the smallest element ⊥ is prime in *M*.

**Theorem 1.** *Assume that* (*<sup>V</sup>*, ⊕, ◦, *F*) *is an M-hazy vector space over an M-hazy field* (*<sup>F</sup>*, +, •)*. Then,* ∀*u* ∈ *V and* ∀*a* ∈ *F:*


**Proof.** We only prove (1). Assume (*a* ◦ *o*)(*o*) = ⊥. By (MHV1) ∀*a* ∈ *F* and *o* ∈ *V*, we have

$$((a \diamond o)(p) \diamond (o \oplus o)(o) \diamond (a \diamond o)(r) \le \bigwedge\_{s \in V} ((a \diamond o)(s) \leftrightarrow (p \oplus r)(s)).$$

When *r* = *s* = *p*, we have,

$$(a \circ o)(p) \diamond (a \circ o)(p) \diamond (a \circ o)(p) \le (p \oplus p)(p).$$

If *p* = *o*, then (*p* ⊕ *p*)(*p*)=(*p* ⊕ (−*p*))(*p*) = ⊥ by the condition (1) of Proposition 4, which is a contradiction. Hence, (*a* ◦ *o*)(*o*) = ⊥.
