Fuzzy congruence relation generated by a fuzzy relation in vector spaces

1 Introduction

The systematic generalization of crisp concepts to the fuzzy case has proven to be an important theoretical tool for the development of new methods of reasoning under uncertainty, imprecision, and lack of information. Regarding the generalization level, it is important to note that the definition of fuzzy sets originally presented as mappings with codomain [0 1], was soon replaced by more general structures, for instance, a complete lattice, as in the L-fuzzy sets introduced by Goguen [2]. Katsaras and Lin in [3] introduced the notion of a fuzzy subspace of a vector space and obtained some fundamental results pertaining to this notion. Subsequently, Das [1] redefined a fuzzy subspace with respect to a t-norm and showed that the most of the results established for fuzzy subspaces with respect to the t-norm ’min’ are valid for t-norms of a more general nature. N. P. Mukherjee in [7] introduced and characterized the concepts of fuzzy normal subgroups and fuzzy cosets and showed that the level subgroups of a fuzzy normal subgroup are normal. Similarly, R. Kumar [4] researched that these subject in vector spaces and introduced the concepts of fuzzy vector spaces and fuzzy cosets. In decision making problems and uncertainty, fuzzy set theory, rough set theory, and soft set theory are essential mathematical tools. J. Zhan et al. in [13 –16] introduced and applied a novel concept of soft rough fuzzy sets, which is called Z-soft rough fuzzy set, by this definition we can remove the restrictive condition for full soft set, also Z-soft rough fuzzy sets are more precise than the other soft rough fuzzy sets that before had been introduced and it is a good advantage of this concept. X. Ma et al. [5] showed decision making based on fuzzy sets, soft sets, fuzzy soft sets and rough soft sets. Congruence relations are basic tools for rough sets and approximation spaces in algebraic systems. For instance, R. Moradian et al. [6] studied rough sets by considering congruence relations induced by fuzzy ideals in BCK-algebras, also the concept of fuzzy congruence relation in the fuzzy systems is as important as the concept of congruence relation in the nonfuzzy systems. Hence we study the concept of a on a vector space and characterize its properties. We define an operation ⊕ between two fuzzy relations on a vector space and show that the set of all fuzzy congruence relations on a vector space with the operation ⊕ is a commutative semi-group. We also introduce the concept of a fuzzy coset relation on a vector space and prove that the set of all fuzzy coset relations on a vector space with the operation ⊕ and scalar operation that mentioned in this paper is a vector space. By the concept of the subspace and the subspace generated by a set, we give the construction of a fuzzy congruence relation generated by a fuzzy relation on a vector space. For this purpose, by the concept of a fuzzy equivalence relation we define regular and irregular fuzzy relations and construct a generated by a regular fuzzy relation. Therefore to construct a generated by an irregular fuzzy relation, it is sufficient to convert it into a regular fuzzy relation that this method is shown in this paper.

2 Preliminaries

Let X be a non-empty set. A fuzzy set on X is a function μ : X → [0, 1]. Let μ be a fuzzy set on X. Then for any t ∈ [0, 1], the set μ _t = {x ∈ X | μ (x) ≥ t}, is called a level subset of μ.

Definition 2.1. [10, 11] Let X be a non-empty set. A fuzzy relation on X is a map η : X × X → [0, 1] and R (X) will be denote the set of all fuzzy relations on X.

Definition 2.2. [10] Let φ, ψ ∈ R (X). Then the operations ⊆, ∪, ∩, φ ^-1 and φ ∘ ψ on R (X) are defined as follows:

φ ⊆ ψ if only if φ (x, y) ≤ ψ (x, y),

Definition 2.3. [8, 9] A fuzzy relation ρ on X is called a fuzzy equivalence relation on X, if for all x, y ∈ X,

ρ (x, x) =1,

Definition 2.4. [3] Let V be a vector space on filed F = ℝ and μ be a fuzzy set on V. Then μ is called a fuzzy subspace of V if for all u, v ∈ V and a, b ∈ F, μ ( av + bu ) ≥ μ ( v ) ∧ μ ( u )

Throughout the paper, we assume that V is a vector space over filed F = ℝ , unless otherwise specified.

3 Fuzzy congruence relation on vector spaces

In this section, we introduce the concept of a fuzzy congruence relation on a vector space and try to illustrate this subject by some examples and theorems.

Definition 3.1. A fuzzy equivalence relation ρ on V is called a fuzzy congruence relation on V if for all x, y, z, t ∈ V and r, s ∈ F, ρ ( rx + z , sy + t ) ≥ ρ ( x , y ) ∧ ρ ( z , t )

Example 3.2. If V = ℤ 4 and F = ℤ 2 , then V is a vector space on F. New we define the fuzzy relation ρ on V by: ρ ( x , y ) = { 1 if x = y 0.5 if ( x , y ) ∈ { ( 2 , 3 ) , ( 3 , 2 ) } 0.3 otherwise Then ρ is a fuzzy equivalence relation on V, but it is not a fuzzy congruence relation on V, since ρ (2 +2, 3 + 2) ngeqslantρ (2, 3) ∧ ρ (2, 2).

Lemma 3.3. Let ρ be a fuzzy congruence relation on V. Then for any x, y, z ∈ X:

ρ (rx, sy) ≥ ρ (x, y), for all r, s ∈ F and ρ (rx, sy) = ρ (x, y), for all r, s ∈ F - {0}.

Proof. By the definition of fuzzy congruence relation on V, for any r, s ∈ F, we have ρ (rx, sy) = ρ (rx + 0, sy + 0) ≥ ρ (x, y) ∧ ρ (0, 0) = ρ (x, y) If r, s ∈ F - {0}, then by the first part, we have ρ (x, y) = ρ (r ^-1 rx, s ^-1 sy) ≥ ρ (rx, sy). Therefore ρ (rx, sy) = ρ (x, y).

(ii) Let r, s ∈ F - {0} and ρ (x, y) < ρ (z, t). It is clear that ρ (x, y) ≤ ρ (rx + z, sy + t). On the other hand, since ρ (x, y) = ρ (rx + z - z, sy + t - t) ≥ ρ (- z, - t) ∧ ρ (rx + z, sy + t), we have ρ (x, y) ≥ ρ (rx + z, sy + t) and it completes the proof.

Theorem 3.4. Let α be a fuzzy relation on V. Then α is a fuzzy congruence relation on V if only if for all t ∈ [0, 1], α _t = {x - y ∈ V | α (x, y) ≥ t} is a subspace of V.

Proof. Suppose α is a fuzzy congruence relation on V, c ∈ F and β, γ ∈ α _t . It is sufficient to prove that cβ + γ ∈ α _t . By the definition of α _t , there exist x ₁, x ₂, y ₁, y ₂ ∈ V such that β = x ₁ - y ₁ and γ = x ₂ - y ₂. It is clear that cβ + γ = c (x ₁ - y ₁) + x ₂ - y ₂ = (cx ₁ + x ₂) - (cy ₁ + y ₂) and on the other hand since α is a fuzzy congruence relation on V, we have α ( cx 1 + x 2 , cy 1 + y 2 ) ≥ α ( x 1 , y 1 ) ∧ α ( x 2 , y 2 ) ≥ t Hence cβ + γ ∈ α _t . Conversely, if for all t ∈ [0, 1], α _t is a subspace of V, then we prove that α is a fuzzy congruence relation on V. Since α ₁ is subspace of V, for all x ∈ V, we have x - x = 0 ∈ α ₁. Hence α (x, x) ≥1, which implies that α (x, x) =1, so α is reflexive. If α (x, y) = t ₀, then x - y ∈ α _{t 0} , hence - (x - y) = y - x ∈ α _{t 0} , which implies that α (y, x) ≥ α (x, y) = t ₀. Similarly we have α (x, y) ≥ α (y, x). Therefore, α (x, y) = α (y, x), and so α is symmetric. Suppose that α (x, y) = t ₀, α (y, z) = t ₁ and t ₁ ≤ t ₀. Then by the definition of α _t we have x - y ∈ α _{t 0} , y - z ∈ α _{t 1} and α _{t 0} ⊆ α _{t 1} . Hence x - y ∈ α _{t 1} and we have x - z = (x - y) + (y - z) ∈ α _{t 1} . Therefore α (x, z) ≥ α (x, y) ∧ α (y, z) = t ₁, and so α is transitive. Now we prove that α is a fuzzy congruence relation on V. For this, we must show that ρ (rx + z, sy + t) ≥ ρ (x, y) ∧ ρ (z, t) for all x, y, z, t ∈ V and r, s ∈ F. If α (x, y) = t ₀, α (t, z) = t ₁ and t ₁ ≤ t ₀, then we have α _{t 0} ⊆ α _{t 1} , x - y ∈ α _{t 0} and t - z ∈ α _{t 0} . Therefore, r (x - y) = rx - ry, s (x - y) = sx - sy ∈ α _{t 1} and so we have (rx - sy) + (sx - ry) ∈ α _{t 1} . Since α _{t 1} is a subspace of V, there exists a basis {α ₁, α ₂, α _n } for α _{t 1} . On the other hand since (rx - sy) + (sx - ry) = (rx - ry) + (sx - sy) ∈ α _{t 1} , hence rx - sy and sx - ry must be linear combination of α ₁, α ₂, …, α _n , which implies that rx - sy ∈ α _{t 1} . Then (rx + t) - (sy + z) = (rx - sy) + (t - z) ∈ α _{t 1} . Therefore, by the definition of α _{t 1} we have α (rx + t, sy + z) ≥ α (x, y) ∧ α (t, z) = t ₁.

Example 3.5. Let W be a subspace of V, where W ≠ V, and t ∈ [0, 1). Then ρ ( x , y ) = { 1 if x − y ∈ W t otherwise is a fuzzy congruence relation on V. Moreover, ρ ₁ = W and ρ _t = V are subspaces of V.

Definition 3.6. Let α and β be two fuzzy relations on V. Then the composition α ⊕ β, for any a, b, c, d ∈ V and r, s, k, l ∈ F, is defined as follows: ( α ⊕ β ) ( x , y ) = ∨ x = r a + s c y = k b + l d ( α ( a , b ) ∧ β ( c , d ) )

It is clear that α ⊕ β is a fuzzy relation on V. Since operation + is associative and commutative so by Definition 3.6, we have α ⊕ (β ⊕ γ) = (α ⊕ β) ⊕ γ and α ⊕ β = β ⊕ α, for any α, β, γ ∈ R (V). Hence we have the following theorem.

Theorem 3.7. Let FR (V) be the family of all fuzzy relations on V. Then (FR (V), ⊕) is a commutative semigroup.

Theorem 3.8. Let FC (V) be the family of all fuzzy congruence relations on V. Then (FC (V), ⊕) is a commutative semigroup, too.

Proof. Suppose that α and β are fuzzy congruence relations on V. Then by Theorem 3.7, it is sufficient to prove that α ⊕ β is a on V. Let x ∈ V. Then

( α ⊕ β ) ( x , y ) = ∨ x = r a + s c y = k b + l d ( α ( a , b ) ∧ β ( c , d ) ) ≥ ∨ x = r a + s c x = r a + s c { α ( a , b ) ∧ β ( c , c ) } = 1 Hence α ⊕ β, is reflexive. Now let x, y ∈ V. Then ( α ⊕ β ) ( x , y ) = ∨ x = r a + s c y = k b + l d { α ( a , b ) ∧ β ( c , d ) } = ∨ x = r a + s c y = k b + l d { α ( b , a ) ∧ β ( d , c ) } = ( α ⊕ β ) ( y , x ) Hence α ⊕ β, is symmetric. Now, since α and β are transitive, for any x, y, z ∈ V, we have: ( α ⊕ β ) ( x , z ) ∧ ( α ⊕ β ) ( z , y ) = ( ∨ x = r a + s c z = k b + l d { α ( a , b ) ∧ β ( c , d ) } ) ∧ ( ∨ z = k b + l d y = n e + m f { α ( b , e ) ∧ β ( d , f ) } ) = ∨ x = r a + s c , z = k b + l d , y = n e + m f { α ( a , b ) ∧ β ( c , d ) ∧ α ( b , e ) ∧ β ( d , f ) } = ∨ x = r a + s c , z = k b + l d , y = n e + m f { α ( a , b ) ∧ α ( b , e ) ∧ β ( c , d ) ∧ β ( d , f ) } ≤ ∨ x = r a + s c y = n e + m f { α ( a , e ) ∧ β ( c , f ) } = ( α ⊕ β ) ( x , y )

Thus α ⊕ β is transitive. Therefore, α ⊕ β is an equivalence relation. Now, suppose that x, y, t, z ∈ V and m, n ∈ F. Then there exist a ₁, a ₂, b ₁, b ₂, c ₁, c ₂, d ₁, d ₂ ∈ V such that x = a ₁ + c ₁, y = b ₁ + d ₁, t = a ₂ + c ₂ and z = b ₂ + d ₂, which a = a ₁ + a ₂, b = b ₁ + b ₂, c = c ₁ + c ₂ and d = d ₁ + d ₂. Consequently we have ( α ⊕ β ) ( m x + t , n y + z ) = ∨ m x + t = r a + s c n y + z = k b + l d { α ( a , b ) ∧ β ( c , d ) } = ∨ m x + t = ( r a 1 + s c 1 ) + ( r a 2 + s c 2 ) n y + z = ( k b 1 + l d 1 ) + ( k b 2 + l d 2 ) { α ( a 1 + a 2 , b 1 + b 2 ) ∧ β ( c 1 + c 2 , d 1 + d 2 ) } = ∨ m x + t = r ( a 1 + a 2 ) + s ( c 1 + c 2 ) n y + z = k ( b 1 + b 2 ) + l ( d 1 + d 2 ) { α ( a 1 + a 2 , b 1 + b 2 ) ∧ β ( c 1 + c 2 , d 1 + d 2 ) } ≥ ∨ m x + t = r ( a 1 + a 2 ) + s ( c 1 + c 2 ) n y + z = k ( b 1 + b 2 ) + l ( d 1 + d 2 ) { α ( a 1 , b 1 ) ∧ β ( ( c 1 , d 1 ) ∧ α ( a 2 , b 2 ) ∧ β ( c 2 , d 2 ) } = ∨ m x = r a 1 + s c 1 n y = k b 1 + l d 1 x = m − 1 r a 1 + m − 1 s c 1 y = n − 1 k b 1 + n − 1 l d 1 { α ( a 1 , b 1 ) ∧ β ( c 1 , d 1 ) } ∧ ∨ t = r a 2 + s c 2 z = k b 2 + l d 2 { α ( a 2 , b 2 ) ∧ β ( c 2 , d 2 ) } = ∨ x = r ′ a 1 + s ′ c 1 y = k ′ b 1 + l ′ d 1 { α ( a 1 , b 1 ) ∧ β ( c 1 , d 1 ) } ∧ ∨ t = r a 2 + s c 2 z = k b 2 + l d 2 { α ( a 2 , b 2 ) ∧ β ( c 2 , d 2 ) } = ( α ⊕ β ) ( x , y ) ∧ ( α ⊕ β ) ( t , z ) Therefore, by Definition 3.1, α ⊕ β is a on V.

Definition 3.9. Let A, B, C and D be subsets of vector space V, α be a fuzzy relation on V and f and g be homomorphisms from A to B and C to D, respectively. Then,

α is called (f × g)-invariant if (f × g) (x, y) = (f × g) (z, t), then α (x, y) = α (z, t), for any (x, y), (z, t) ∈ A × C, where (f × g) (x, y) = (f (x), g (y)).

Theorem 3.10. Let W be a vector space on F, f and g be homomorphisms from V onto W, f and g be f - g one to one and α be an f × g-invariant on V. Then ( f × g ) ( α ) ( x , y ) = { ∨ α ( a , b ) if ( a , b ) ∈ ( f × g ) − 1 ( x , y ) 0 otherwise is a fuzzy congruence relation on W.

Proof. It is clear that (f × g) (α) is a fuzzy equivalence relation on W. Now we prove that it is a on W. Let x, y, z, t ∈ W and r, s ∈ F. Then ( f × g ) ( α ) ( r x + z , s y + t ) = ∨ ( a , b ) ∈ ( f × g ) − 1 ( r x + z , s y + t ) α ( a , b ) = ∨ f ( a ) = r x + z g ( b ) = s y + t α ( a , b ) = ∨ f ( a ) = f ( r u + v ) u ∈ f − 1 ( x ) , v ∈ f − 1 ( z ) g ( b ) = g ( s w + c ) w ∈ g − 1 ( y ) , c ∈ g − 1 ( t ) α ( a , b ) = ∨ u ∈ f − 1 ( x ) , w ∈ g − 1 ( y ) v ∈ f − 1 ( z ) , c ∈ g − 1 ( t ) α ( r u + v , s w + c ) ≥ ∨ u ∈ f − 1 ( x ) , w ∈ g − 1 ( y ) v ∈ f − 1 ( z ) , c ∈ g − 1 ( t ) α ( u , w ) ∧ α ( v , c ) = ∨ u ∈ f − 1 ( x ) w ∈ g − 1 ( y ) α ( u , w ) ∧ ∨ v ∈ f − 1 ( z ) c ∈ g − 1 ( t ) α ( v , c ) = ( f × g ) ( α ) ( x , y ) ∧ ( f × g ) ( α ) ( z , t )

Corollary 3.11. Let W be a vector space on F. Then,

if f and g are isomorphisms from V to W, f and g are f - g one to one and α is a on V, then (f × g) (α) is a on W.

Proof. The proof is straightforward.

4 Fuzzy coset relations

The concept of a fuzzy coset of a fuzzy subspace introduced by R. Kumar [4]. In this section, we extend this concept to the fuzzy congruence relations in a vector space and introduce the concept of a fuzzy coset relation in a vector space.

Defintion 4.1. Let ρ be a on V. For any a, b ∈ V, the fuzzy relation ρ ^(a,b) on V which is defined by ρ ^(a,b) (x, y) = ρ (a - x, b - y), for all x, y ∈ V, is called a fuzzy coset relation determined by a, b and ρ.

Example 4.2. Let V = F ² and V ₀ = {(α, 0) | α ∈ F}. Then the fuzzy relation ρ on V which is defined by: ρ ( x , y ) = { 1 if x − y ∈ V 0 0.5 if x − y ∈ V − V 0 is a fuzzy congruence relation on V and for a, b ∈ V, fuzzy relation ρ ^(a,b) given by ρ ( a , b ) ( x , y ) = { 1 if ( a - x ) - ( y - b ) ∈ V 0 0.5 if ( a - x ) - ( b - y ) ∈ V - V 0 is a fuzzy coset relation respect to a and b. If we put a = (1, 2) and b = (1, 3), then for x = (2, 1) and y = (3, 1), we have ρ (x, y) =1, while ρ ^(a,b) (x, y) =0.5. Similarly for x = (2, 1) and y = (3, - 1), we have ρ (x, y) = ρ ^(a,b) (x, y) =0.5.

Lemma 4.3. Let ρ ^(a,b) and ρ ^(c,d) be fuzzy coset relations on V. then ρ ( a , b ) = ρ ( c , d ) i f a n d o n l y i f ρ ( a − c , b − d ) = 1.

Proof. If ρ ^(a,b) = ρ ^(c,d), then ρ ^(a,b) (c, d) = ρ (a - c, b - d) = ρ ^(c,d) (c, d) = ρ (c - c, d - d) = ρ (0, 0) =1 Conversely, assume ρ (a - c, b - d) =1. Then we have ρ ^(a,b) (x, y) ≤ ρ (a - c, b - d). If ρ ^(a,b) (x, y) = ρ (a - c, b - d), then ρ (c - x, d - y) = ρ (a - x - (a - c), b - y - (b - d)) ≥ ρ (a - x, b - y) ∧ ρ (a - c, b - d). Hence ρ ^(a,b) (x, y) = ρ ^(c,d) (x, y). Now, if ρ ^(a,b) (x, y) < ρ (a - c, b - d), then by Lemma 3.3(ii), we have ρ ^(a,b) (x, y) = ρ (a - x, b - y) = ρ (a - x - (a - c), b - y - (b - d)) = ρ (c - x, d - y) = ρ ^(c,d) (x, y).

We define the concepts of addition and scalar multiplication on fuzzy coset relations as follows: ( ρ ( a , b ) ⊕ ρ ( c , d ) ( x , y ) ) = ∨ x = x 1 + x 2 y = y 1 + y 2 { ρ ( a , b ) ( x 1 , y 1 ) ∧ ρ ( c , d ) ( x 2 , y 2 ) } k . ρ ( a , b ) = ρ ( k a , k b )

Proposition 4.4. Let ρ ^(a,b) and ρ ^(c,d) be two fuzzy coset relations on V. Then ( ρ ( a , b ) ⊕ ρ ( c , d ) ) ( x , y ) = ρ ( a + c , b + d ) ( x , y )

Proof. First, we prove that the above addition is well-defined. Suppose that ρ ^(a,b) = ρ ^(c,d) and ρ ^(e,f) = ρ ^(g,h). Then by Lemma 4.3, ρ (a + e - (c + g), b + f - (d + h)) ≥ ρ (a - c, b - d) ∧ ρ (e - g, f - h) =1 Therefore, ρ ^(a,b) ⊕ ρ ^(e,f) = ρ ^(c,d) ⊕ ρ ^(g,h). For all x ₁, x ₂, y ₁, y ₂ ∈ V such that x = x ₁ + x ₂ and y = y ₁ + y ₂ we have ρ ( a + c , b + d ) ( x , y ) = ρ ( a - x 1 + c - x 2 , b - y 1 + d - y 2 ) ≥ ρ ( a - x 1 , b - y 1 ) ∧ ρ ( c - x 2 , d - y 2 ) = ρ ( a , b ) ( x 1 , y 1 ) ∧ ρ ( c , d ) ( x 2 , y 2 ) Therefore ρ ( a + c , b + d ) ( x , y ) ≥ ∨ x = x 1 + x 2 y = y 1 + y 2 { ρ ( a , b ) ( x 1 , y 1 ) ∧ ρ ( c , d ) ( x 2 , y 2 ) } = ( ρ ( a , b ) ⊕ ρ ( c , d ) ) ( x , y ) Conversly ρ ( a , c ) ⊕ ρ ( c , d ) ( x , y ) = ∨ x = x 1 + x 2 y = y 1 + y 2 { ρ ( a , b ) ( x 1 , y 1 ) ∧ ρ ( c , d ) ( x 2 , y 2 ) } If (ρ ^(a,b) (x ₁, y ₁) ≠ ρ ^(c,d) (x ₂, y ₂)), then by Lemma 3.3 we have ∨ x = x 1 + x 2 y = y 1 + y 2 { ρ ( a , b ) ( x 1 , y 1 ) ∧ ρ ( c , d ) ( x 2 , y 2 ) } ≥ ∨ x = x 1 + x 2 y = y 1 + y 2 { ρ ( a , b ) ( x 1 , y 1 ) ∧ ρ ( c , d ) ( x 2 , y 2 ) } = ∨ x = x 1 + x 2 y = y 1 + y 2 { ρ ( a + c − ( x 1 + x 2 ) , b + d − ( y 1 + y 2 ) ) = ρ ( a + c , b + d ) ( x , y )

Theorem 4.5. Let (V × V) _ρ = {ρ ^(a,b)|a, b ∈ V}. Then (V × V) _ρ is a vector space over F.

Proof. It is easy to verify that (V × V) _ρ , is a vector space under scalar multiplication and addition operations that before determined.

Theorem 4.6. Let ρ be a on V. Then the map φ : V × V ⟶ (V × V) _ρ which is defined by φ (a, b) = ρ ^(a,b), for all a, b ∈ V, is an onto linear homomorphism with Kerφ = I _V .

Proof. Clearly, φ is an onto linear homomorphism. If (x, y) ∈ Kerφ, then φ (x, y) = ρ ^(x,y) = ρ ^(0,0) and so by Lemma 1, we have ρ (x, y) =1. Therefore (x, y) ∈ I _V , which is complete the proof.

Corollary 4.7. Let ρ be a on V. then

(V × V)/I _V ≅ (V × V) _ρ .

5 Union of fuzzy congruence relations

It is known that the union of two subspaces is a subspace if only if at least one of them is contained in the other one. In Theorems 5.2 and 5.4, we show that this result does not hold in general for fuzzy congruence relations. In Theorem 5.3, we show that if Im (ρ ₁ ∪ ρ ₂) = {0, 1}, then either ρ ₁ ⊆ ρ ₂ or ρ ₂ ⊆ ρ ₁.

Example 5.1. Let W ₁ and W ₂ be two proper subspaces of V, W ₁ ∩ W ₂ = 0 and t _i ∈ [0, 1), where 1 ≤ i ≤ 2, such that 1 > t ₁ > t ₂. Then by Example 3.5, the fuzzy relations ρ 1 ( x , y ) = { 1 if x − y ∈ W 1 t 1 otherwise ρ 2 ( x , y ) = { 1 if x − y ∈ W 2 t 2 otherwise are fuzzy congruence relations on V, but the union of these fuzzy congruence relations given by

ρ ( x , y ) = ( ρ 1 ∪ ρ 2 ) ( x , y ) = { 1 if x − y ∈ W 1 ∪ W 2 t 1 otherwise

is not a on V. Since 0 ≠ α ₁ ∈ W ₁ and 0 ≠ α ₂ ∈ W ₂, but ρ ( α 1 + α 2 , 0 ) = t 1 ≱ ρ ( α 1 , 0 ) ∧ ρ ( α 2 , 0 ) = 1

Theorem 5.2. Let ρ be a fuzzy congruence relation on V such that Imρ = {1, t} and 0 < t < 1. Then there exist fuzzy congruence relations ρ ₁ and ρ ₂ on V such that ρ = ρ ₁ ∪ ρ ₂, ρ ₁ ⊈ ρ ₂ and ρ ₂ ⊈ ρ ₁.

Proof. Let W ₁ and W ₂ be subspaces of V such that W ₁ ⊂ W ₂ and 1 > t ₁ > t ₂ ≥ 0. Then the fuzzy relations ρ ₁ and ρ ₂ on V given by ρ 1 ( x , y ) = { 1 if x − y ∈ W 1 t 1 otherwise ρ 2 ( x , y ) = { 1 if x − y ∈ W 2 t 1 otherwise are fuzzy congruence relations on V, but ρ ₁ ⊈ ρ ₂ and ρ ₂ ⊈ ρ ₁. Also ρ ( x , y ) = ( ρ 1 ∪ ρ 2 ) ( x , y ) = { 1 if x − y ∈ W 2 t 1 otherwise

Theorem 5.3. Let ρ, ρ ₁ and ρ ₂ be fuzzy congruence relations on V such that ρ = ρ ₁ ∪ ρ ₂. If Imρ = {0, 1}, then either ρ ₁ ⊆ ρ ₂ or ρ ₂ ⊆ ρ ₁.

Proof. In order to prove by contradiction we suppose that there exist x, y, z, w ∈ V, such that ρ ₁ (x, y) > ρ ₂ (x, y) and ρ ₁ (z, w) < ρ ₂ (z, w). Since Imρ = {0, 1}, we have ρ (x, y) = ρ ₁ (x, y) =1 and ρ (z, w) = ρ ₂ (z, w) =1. Hence by Lemma 3.3(ii), we have ρ ₂ (x, y) = ρ ₂ (ax + z, ay + w) =0 and ρ ₁ (z, w) = ρ ₁ (ax + z, ay + w) =0 and so ρ (ax + z, ay + w) = max {ρ ₁ (ax + z, ay + w), ρ ₂ (ax + z, ay + w)} =0 Now, because ρ is a congruence relation, then ρ (ax + z, ay + w) ≥ ρ (x, y) ∧ ρ (z, w) =1, that is a contradiction.

Theorem 5.4. Let ρ be a on V such that 3≤ |Imρ| ≤ ∞. Then there exist fuzzy congruence relations ρ ₁ and ρ ₂ such that ρ = ρ ₁ ∪ ρ ₂, ρ ₁ ⊈ ρ ₂ and ρ ₂ ⊈ ρ ₁.

Proof. Let ρ be a on V and Imρ = {t ₀ = 1, t ₁, …, t _n }, which 2≤ n ≤ ∞ and 1 = t ₀ > t ₁ > t ₂ > … > t _n . Then we choose r ₁, r ₂ ∈ [0, 1] such that 1 = t ₀ > t ₁ > r ₁ > t ₂ > r ₂ > t ₃ > t _n ρ 1 ( x , y ) = { 1 if x − y ∈ ρ t 0 t 1 if x − y ∈ ρ t 1 − ρ t 0 r 2 if x − y ∈ ρ t 2 − ρ t 1 ρ ( x , y ) otherwise ρ 2 ( x , y ) = { 1 if x − y ∈ ρ t 0 r 1 if x − y ∈ ρ t 1 − ρ t 0 t 2 if x − y ∈ ρ t 2 − ρ t 1 ρ ( x , y ) otherwise It is obvious that ρ ₁ and ρ ₂ are fuzzy congruence relations on V and ρ = ρ ₁ ∪ ρ ₂, but ρ ₁ ⊈ ρ ₂ and ρ ₂ ⊈ ρ ₁

6 Fuzzy congruence relations generated by a fuzzy relations

In this section, we give a construction of the fuzzy congruence relation generated by a fuzzy relation in a vector space and characterize it. For this purpose we define regular and irregular fuzzy relations. The generated by a regular fuzzy relation ρ and the vector space generated by a set A is denoted by 〈ρ〉 and respectively.

Definition 6.1. Let α be a fuzzy relation on V. Then α is called a regular fuzzy relation if there exists a fuzzy equivalence relation ρ on V such that for all (x, y) ∈ Dom (α), α (x, y) = ρ (x, y). A fuzzy relation α is called an irregular fuzzy relation if it is not a regular fuzzy relation.

Example 6.2. Let W be a vector space and S = {o, a, b, c, d} be a subset of V. Then we define the fuzzy relations β, θ and γ on W as follows: β ( u , v ) = { 0.5 if u = v = a or u = v = c 0.3 if u , v ∈ { d , b } and u ≠ v 0.2 if u , v ∈ { b , c , o } and u ≠ v 0 otherwise θ ( u , v ) = { 0.3 if u = a and v = b 0.2 if u = b and v = a 0.1 if u , v ∈ { b , c , o } and u ≠ v 0 otherwise γ ( u , v ) = { 0.3 if u = a and v = b 0.2 if u = b and v = c 0.1 if u , v ∈ { a , c , o } and u ≠ v 0 otherwise Since for any fuzzy equivalence relation ρ, we have ρ (a, a) = ρ (b, b) =1 and on the other hand β (a, a) = β (c, c) =0.5 < 1, hence there is not a fuzzy equivalence relation ρ such that β (a, a) = ρ (a, a) and β (c, c) = ρ (b, b). Therefore β is not regular and is an irregular fuzzy relation and in this case, we say that β contradicts the reflexivity property for a and b. Similarly since θ (a, b) ≠ θ (b, a) and any fuzzy equivalence relation is symmetric, hence θ is not regular and in this case, we say that θ contradicts the symmetry property for a and b. Since γ (a, b) ∧ γ (b, c) =0.2 > γ (a, c) =0.1, and any fuzzy equivalence relation is transitive, hence there is not a fuzzy equivalence relation ρ such that γ (a, b) = ρ (a, b), γ (b, c) = ρ (b, c) and γ (a, c) = ρ (a, c). Consequently γ is not regular fuzzy relation and in this case, we say that γ contradicts the transitivity property for a, b and c.

Theorem 6.3. Let ρ be a regular fuzzy relation on V, Imρ = {t ₀, t ₁, …., t _n }, such that 1 ≥ t ₀ > t ₁ > t ₂ …. > t _n ≥ 0, A _i = {x, y ∈ V|ρ (x, y) = t _i and x ≠ y} and for 0 ≤ i ≤ n, where W _-1 =∅. Hence: if t ₀ = 1 and W _n = V, then 〈 ρ 〉 ( x , y ) = { 1 if x - y ∈ W 0 t i if x - y ∈ W i - W i - 1 1 ≤ i ≤ n if t ₀ ≠ 1 and W _n = V, then 〈 ρ 〉 ( x , y ) = { 1 if x - y ∈ 〈 0 〉 t 0 if x - y ∈ W 0 - 〈 0 〉 t i if x - y ∈ W i - W i - 1 1 ≤ i ≤ n if t ₀ = 1 and W _n ⊂ V, then 〈 ρ 〉 ( x , y ) = { 1 if x - y ∈ W 0 t i if x - y ∈ W i - W i - 1 1 ≤ i ≤ n 0 if x - y ∈ V - W n if t ₀ ≠ 1 and W _n ⊂ V, then 〈 ρ 〉 ( x , y ) = { 1 if x - y ∈ 〈 0 〉 t 0 if x - y ∈ W 0 - 〈 0 〉 t i if x - y ∈ W i - W i - 1 1 ≤ i ≤ n 0 if x - y ∈ V - W n

Proof. Suppose that ρ is a regular fuzzy relation on V. Then it is clear that 〈ρ〉 is reflexive and symmetric. Now we prove that 〈ρ 〉 (x, y) ∧ 〈 ρ 〉 (y, z) ≤ 〈 ρ 〉 (x, z). If 〈ρ 〉 (x, y) = t _i and 〈ρ 〉 (y, z) = t _j , then we have x - y ∈ W _i - W _i-1 and y - z ∈ W _j - W _j-1. We assume that t _i > t _j and α ₁, α ₂, … α _p , α _q , is a basis for W _i , which α ₁, α ₂, … α _p is a basis for W _i-1. Then W _i - W _i-1 = {c ₁ α ₁ + … + c _p α _p + …. + c _q α _q |c _p+1 ≠ 0 or c _p+2 ≠ 0 … or c _q ≠ 0}. Similarly W _j - W _j-1 = {a ₁ α ₁ + … + a _q α _q + …. + a _r α _r , …. α _s |a _r+1 ≠ 0 or a _r+2 ≠ 0 … or a _s ≠ 0} which α ₁, α _{i 2} , … α _{i r} is a basis for W _J-1 and α ₁, α ₂, … α _s is a basis for W _J . Since t _i ≥ t _j , we have W _i ⊆ W _j and W _i-1 ⊆ W _j-1. Hence x - y + y - z = x - z ∈ W _i - W _i-1 + W _j - W _j-1 = W _j - W _j-1. Therefore, 〈ρ 〉 (x, y) ∧ 〈 ρ 〉 (y, z) = t _j = 〈 ρ 〉 (x, z) and so 〈ρ〉 is transitive. Now we prove that 〈ρ 〉 (ax + z, by + w) ≥ 〈 ρ 〉 (x, y) ∧ 〈 ρ 〉 (z, w). Similar to the transitively we assume that 〈ρ 〉 (x, y) = t _i , 〈ρ 〉 (z, w) = t _j and t _i ≥ t _j . Then x - y = ∑_∈Λ1 c _i α _i ∈ W _i - W _i-1, where x = ∑ _i∈Λ11 c _i α _i , y = ∑ _i∈Λ12 c _i α _i and Λ₁₁ ∪ Λ₁₂ = Λ₁. Hence for any a, b ∈ F, ax - by = ∑_∈Λ1 s _i α _i ∈ W _i - W _i-1 and z - w = ∑_∈Λ2 d _j α _j ∈ W _j - W _j-1. Therefore, (ax + z) - (by + w) = (ax - by) + (z - w) ∈ W _j - W _j-1. So 〈ρ 〉 (ax + z, by + w) = t _j = 〈 ρ 〉 (x, y) ∧ 〈 ρ 〉 (z, w). Thus 〈ρ〉 is a on V. It is sufficient to prove that 〈ρ〉 is the least on V. For this we must show that if θ is a on V containing ρ, then θ (x, y) ≥ 〈 ρ 〉 (x, y). If t ₀ = 1 and W _n = V, then for x, y ∈ V there exists 1 ≤ i ≤ n such that x - y ∈ W _i - W _i-1 ⊆ W _i . On the other hand we have, such that A _k = {a _kj , b _kj ∈ V|ρ (a _kj , b _kj ) = t _k , a _kj ≠ b _kj }, where 0 ≤ k ≤ i. Therefore, x - y and consequently x and y are linear combinations of the elements in {A ₀, A ₁, …, A _i }. Hence we have x = ∑ _j∈Λ1 (c _kj a _kj + d _kj b _kj ) and y = ∑_∈Λ2 (e _kj a _kj + g _kj b _kj ), 0 ≤ k ≤ i. Thus

θ ( x , y ) = θ ( ∑ j ∈ Λ 1 ( c k j a k j + d k j b k j ) , ∑ j ∈ Λ 2 ( e k j a k j + g k j b k j ) ) ≥ θ ( ∑ j ∈ Λ 1 ( c k j a k j ) , ∑ j ∈ Λ 1 ( g k j b k j ) ) ∧ θ ( ∑ j ∈ Λ 1 ( d k j b k j ) , ∑ j ∈ Λ 2 ( e k j a k j ) ) ≥ ∧ { θ ( a 0 j , b 0 j ) , … , θ ( a i j , b i j ) } ( j ∈ Λ 1 ∪ Λ 1 ) ≥ ∧ { ρ ( a 0 j , b 0 j ) , … , ρ ( a i j , b i j ) } ( j ∈ Λ 1 ∪ Λ 2 ) = t i = 〈 ρ 〉 ( x , y ) .

If t ₀ ≠ 1 and W _n = V, then θ (x, x) =1 = 〈 ρ 〉 (x, x). Similar to the first state for x ≠ y ∈ V, we have θ (x, y) ≥ 〈 ρ 〉 (x, y). If W _n ⊂ V, then for x, y ∈ V, there exists 0 ≤ i ≤ n, such that x - y ∈ W _i - W _i-1 or x - y ∈ V - W _n . If x - y ∈ W _i - W _i-1, then similar to the last states θ (x, y) ≥ 〈 ρ 〉 (x, y) else if x - y ∈ V - W _n , then it is clear that θ (x, y) ≥ 〈 ρ 〉 (x, y) =0.

Example 6.4. Let W be the vector space of polynomial functions from ℝ into ℝ which have degree less than or equal to 4 and let {1, t, t ², t ³, t ⁴} be a basis for W, 1 = t ₀ > t ₁ > t ₂ ≥ 0, and ρ be a fuzzy relation on W as follows: ρ ( u , v ) = { 1 if u , v ∈ { 1 + t , 2 t } t 1 if u = t 3 + t , v = 2 t 2 if u ≠ v , u , v ∈ { t 2 + 1 , t 4 } 0 otherwise It is clear that ρ is a regular fuzzy relation and by Theorem 6.3, we have

〈 ρ 〉 ( x , y ) = { 1 if x − y ∈ 〈 1 + t , 2 t 〉 = 〈 1 , t 〉 t 1 if x − y ∈ 〈 1 , t , t 3 + t , 2 〉 − 〈 1 , t 〉 t 2 if x − y ∈ 〈 1 , t , t 2 , t 3 , t 4 〉 − 〈 t 2 + 1 , t 4 〉

Note 6.5. We can convert an irregular fuzzy relation into a regular fuzzy relation as follows: Let α be an irregular fuzzy relation on V and {o, a, b, c, d} ⊆ V. Then we have at least one of these cases:

Case 1: α contradicts only the reflexive property for some elements in its domain. In this case to convert α into the regular fuzzy relation, we must delete this contradiction, for example it is easy to check that the fuzzy relation β given by β ( u , v ) = { 0.5 if u , v ∈ { a , c } 0.3 if u , v ∈ { d , b } and u ≠ v 0.2 if u , v ∈ { b , c , o } and u ≠ v 0 otherwise is an irregular fuzzy relation and contradicts the reflexive property for a and c, since β (a, a) = β (c, c) =.5 < 1. If we omit the contradiction that exists for the reflexivity property for a and c, then we have β ¯ ( u , v ) = { 0.5 if u , v ∈ { a , c } and u ≠ v 0.3 if u , v ∈ { d , b } and u ≠ v 0.2 if u , v ∈ { b , c , o } and u ≠ v 0 otherwise Clearly β ¯ is a regular fuzzy relation, β = β ¯ expect in (a, a) and (c, c), but β ( a , a ) = β ( c , c ) = . 5 < 〈 β ¯ 〉 ( a , a ) = 〈 β ¯ 〉 ( a , a ) = 1 , hence β ⊆ 〈 β ¯ 〉 and consequently we have 〈 β ¯ 〉 = 〈 β 〉 .