Construction of some algebras of logic by using fuzzy ideals in mv-modules

3.1 Fuzzy A-ideals

Let σ ∈ [0, 1] ^M and consider the following properties:

(FI1)  σ (0) ≥ σ (x).

(FI2)  σ (y) ≥ σ (x) ⊓ σ (y x^★).

(FI3)  σ (αx) ≥ σ (x), for α ∈ A.

(FI4)  σ (a b) ≥ σ (a) ⊓ σ (b).

(FI5)  a ≼ b ⇒ σ (a) ≥ σ (b).

(FI6)  z^★ y x = 0^★ ⇒ σ (z) ≥ σ (x) ⊓ σ (y).

(FI7)  z ≼ x y ⇒ σ (z) ≥ σ (x) ⊓ σ (y).

According to [15], a fuzzy A-ideal of mv-module M is a fuzzy subset of M satisfying (FI1), (FI2), and (FI3). The set of all fuzzy A-ideals of M will be denoted by fai (M). Theorem 3.1 gives some equivalent conditions for a fuzzy A-ideal.

Theorem 3.1. Let σ ∈ [0, 1] ^M. Then, the following are equivalent:

σ ∈ fai (M).

Proof. (1) Ra (2) Assume that a ≼ b, for a, b ∈ M. Then a^★ b = 0^★ and so a b^★ = 0. Now, by (FI2) and then (FI1) we have σ ( a ) ≥ σ ( b ) ⊓ σ ( a b ★ ) = σ ( b ) ⊓ σ ( 0 ) = σ ( b ) , proving (FI5). To prove (FI4), taking x = a and y = a b in (FI2) and then using (FI5) we get σ ( a b ) ≥ σ ( a ) ⊓ σ ( ( a b ) a ★ ) = σ ( a ) ⊓ σ ( a ★ ⊓ b ) ≥ σ ( a ) ⊓ σ ( b ) .

(2) Ra (3) Let x, y, z ∈ M such that z^★ y x = 0^★. Then z ≼ y x and so by (FI5) and then (FI4) we get that σ (z) ≥ σ (y x) ≥ σ (y) ⊓ σ (x), proving (FI6).

(3) ⇔ (4) Straightforward.

(4) Ra (1) Let x, y ∈ M. From y ≼ x ⊔ y = x ( y x ★ ) , by (FI7) we get that σ (y) ≥ σ (x) ⊓ σ (y x^★), proving (FI2). Taking z = 0 and y = x, in (FI7), we get σ (0) ≥ σ (x), proving (FI1). Hence, σ ∈ fai (M).□

Notation. The functions x ↦ 0 and x ↦ 1, which are obviously fuzzy A-ideals of M will be denoted by zero and unit, respectively.

Example 3.2. [15] Let M = A = 2^{1,2}. Then A is a pmv-algebra, where  : =∪, • : =∩ and M is an mv-module over A together with (α, X) ↦ α ∩ X. We define the mapping σ by σ (∅) = t and for all x ∈ M \ {∅}, σ (x) = s, where 0 ≤ s < t < 1. It is routine to verify that σ is a fuzzy A-ideal of M.

Example 3.3. Let ℝ be the real line, M₂ (ℝ) the ring of 2-square matrices over ℝ, zero the zero matrix, and v = ( 1 / 2 1 / 2 1 / 2 1 / 2 ) . If we define the ordering relation on M₂ (ℝ) as X = ( x ij ) i , j = 1 , 2 ≥ 0 LR x ij ≥ 0 , then A = Γ (M₂ (ℝ) , v) is a pmv-algebra and M = Γ (ℝ², u), where u = (1, 1) is an A-module (see [11]). Let σ be a fuzzy subset of M defined by μ ((0, 0)) = t and μ ((x, y)) = s, for all (x, y) ∈ ℝ² \ {(0, 0)}, where 0 ≤ s < t ≤ 1. Obviously, μ is a fuzzy A-ideal of M. Moreover, every non-constant fuzzy ideal of M is of the given form. To see this, assume that a, b ∈ ℝ be such that at least one of them differ from 0. For convenience, assume that a ≠ 0. From (FI3) we have σ ( ( 1 0 ) ) = σ ( ( 1 / a 0 0 0 ) ( a 0 ) ) ≥ σ ( ( a 0 ) ) Similarly, we can deduce that σ ( ( a 0 ) ) ≥ σ ( ( 1 0 ) ) Moreover, if b also differ from 0, we can deduce that σ ( ( a b ) ) = σ ( ( 1 1 ) ) Therefore, for any (a, b) , (x, y) ∈ ℝ² \ {(0, 0)} we must have σ ((a, b)) = σ ((x, y)). Considering (FI1), we deduce that if σ is a non-constant fuzzy A-ideal of M, σ ((0, 0)) > σ ((a, b)), for all (a, b) ∈ ℝ² \ {(0, 0)} and so σ must be two-valued.

Remark 3.4. We observe that Faid(M) is closed with respect to the arbitrary intersections. Hence, for Σ, a family of fuzzy A-ideals of M, the least fuzzy A-ideal of M containing Σ exists; we call it the fuzzy A-ideal generated by Σ, denoted by Fg (∪ Σ). As usual, when Σ = {σ} is singleton we just write Fg (σ). Forouzesh [15] proved that the fuzzy A-ideal generated by σ ∈ [0, 1] ^M is characterized as

〈 σ 〉 ( x ) = ⨆ { σ ( i 1 ) ⊓ ⋯ ⊓ σ ( i r ) ⊓ σ ( a i + 1 i r + 1 ) ⊓ ⋯ σ ( a k i k ) : x ≼ i 1 ⋯ i r a r + 1 i r + 1 ⋯ a k i k , k ∈ ℕ , i 1 , … , i k ∈ M , a r + 1 , … , a k ∈ A }(3.1) Another characterization of Fg (σ) is as follows.

Proposition 3.5. Let σ ∈ [0, 1] ^M. Then

Fg ( σ ) ( x ) = ⨆ { σ ( i 1 ) ⊓ ⋯ ⊓ σ ( i r ) : x ≼ i 1 ⋯ i r , r ∈ ℕ , i 1 , … , i r ∈ M }(3.2) is the least fuzzy A-ideal of M which contains σ _α’s. If M is unital, (3.1) and (3.2) are equivalent.

Proof. We shall prove that Fg (σ) satisfies (FI3), (FI4) and (FI5). For x, y ∈ M and ɛ > 0 arbitrary, from x ≼ x, we always can find n ∈ ℕ and i₁, i₂, …, i _n  ∈ M such that x ≼ i 1 i 2 ⋯ i n(3.3) and Fg ( σ ) ( x ) < ɛ + ( σ ) ( i 1 ) ⊓ ( σ ) ( i 2 ) ⊓ ⋯ ⊓ ( σ ) ( i n )(3.4) Similarly, there exists m ∈ ℕ and j₁, j₂, …, j _m  ∈ M such that y ≼ j 1 j 2 ⋯ j m(3.5) and Fg ( σ ) ( y ) < ɛ + ( σ ) ( j 1 ) ⊓ ( σ ) ( j 2 ) ⊓ ⋯ ⊓ ( σ ) ( j m )(3.6) Now, (3.3) and (3.5) imply that x y ≼ i 1 ⋯ i n j 1 ⋯ j m(3.7) Combining (3.4), (3.6) and (3.7) we get that Fg ( σ ) ( x y ) = ⨆ { σ ( z 1 ) ⊓ ⋯ ⊓ σ ( z r ) : x y ≼ z 1 ⋯ z r , r ∈ ℕ , z 1 , … , z r ∈ M } ≥ σ ( i 1 ) ⊓ ⋯ ⊓ σ ( i n ) ⊓ σ ( j 1 ) ⊓ ⋯ ⊓ σ ( j m ) > ( Fg ( σ ) ( x ) - ɛ ) ⊓ ( Fg ( σ ) ( y ) - ɛ ) = Fg ( σ ) ( x ) ⊓ Fg ( σ ) ( y ) - ɛ , whence Fg (σ) (x y) ≥ Fg (σ) (x) ⊓ Fg (σ) (y), proving (FI4).

To prove (FI5), let x ≼ y. Similar to the proof of the previous step we have Fg ( σ ) ( y ) < ɛ + σ ( i 1 ) ⊓ σ ( i 2 ) ⊓ ⋯ ⊓ σ ( i n )(3.8) for some n ∈ ℕ and i₁, i₂, …, i _n  ∈ M such that y ≼ i₁ i₂  ⋯ i _n . From x ≼ y it follows that x ≼ i 1 i 2 ⋯ i n(3.9) and combining (3.8) and (3.9) we get Fg ( σ ) ( x ) = ⨆ { σ ( j 1 ) ⊓ σ ( j 2 ) ⊓ ⋯ ⊓ σ ( j r ) : x ≼ j 1 j 2 ⋯ j r , r ∈ ℕ , j 1 , … , j r ∈ M } ≥ σ ( i 1 ) ⊓ σ ( i 2 ) ⊓ ⋯ ⊓ ∪ σ ( i n ) > Fg ( σ ) ( y ) - ɛ proving Fg (σ) (x) ≥ Fg (σ) (y).

To prove (FI3), let x ∈ M, a ∈ A and ɛ > 0. Similar to the first part of the proof we can find a natural number n and i₁, i₂, …, i _n  ∈ M such that x ≼ i₁ i₂  ⋯ i _n and Fg ( x ) < ɛ + σ ( i 1 ) ⊓ σ ( i 2 ) ⊓ ⋯ ⊓ σ ( i n ) From Proposition 2 it follows that ax ≼ ai 1 a i 2 ⋯ ai n , whence we conclude that Fg ( σ ) ( ax ) ≥ σ ( ai 1 ) ⊓ σ ( ai 2 ) ⊓ ⋯ ⊓ σ ( ai n ) ≥ σ ( i 1 ) ⊓ σ ( i 2 ) ⊓ ⋯ ⊓ σ ( i n ) > Fg ( σ ) ( x ) - ɛ Since ɛ is arbitrary it follows that Fg (σ) (ax) ≥ Fg (σ) (x). Thus, Fg (σ) is a fuzzy A-ideal of M. Now, let β be a fuzzy A-ideal of M containing σ. Then Fg ( σ ) ( x ) = ⨆ { σ ( i 1 ) ⊓ σ ( i 2 ) ⊓ ⋯ ⊓ σ ( i r ) : x ≼ i 1 i 2 ⋯ i r , r ∈ ℕ , i 1 , . . . , i r ∈ M } ≤ ⨆ { β ( i 1 ) ⊓ β ( i 2 ) ⊓ ⋯ ⊓ β ( i r ) : x ≼ i 1 i 2 ⋯ i r , r ∈ ℕ , i 1 , … , i r ∈ M } ≤ ⨆ { β ( i 1 i 2 ⋯ i r ) : x ≼ i 1 i 2 ⋯ i r , r ∈ ℕ , i 1 , . . . , i r ∈ M } ≤ β ( x ) , proving that Fg (σ) is the least fuzzy A-ideal of M containing σ.

For the last part, it is enough we observe that if M is unital so every element x of M can be written as ax, for some a ∈ A. Hence the inequality x ≼ i 1 ⋯ i r . holds if and only if the inequality x ≼ i 1 ⋯ i k a k + 1 x k + 1 ⋯ a r x r . holds, where a _j  ∈ A and x _j  ∈ M, for j = k + 1, …, r.□

Theorem 3.6. The mapping C : 2^FI(M)ra2^FI(M) defined by C (A) = Fg (∪ A) is an algebraic closure operator.

Proof. Let X⊂FI (M). Then X⊂ ⋃ X⊂Fg (∪ X) = C (X). Now, let X = { σ α : α ∈ I , σ α ∈ FI ( M ) } Y = { σ β : β ∈ J , σ β ∈ FI ( M ) } and X⊂Y. Also, let x ≼ i₁ i₂  ⋯ i _r , where r ∈ ℕ and i₁, . . . , i _r  ∈ X. Obviously, C ( X ) ( x ) = ⨆ { ( ∪ σ β ) ( i 1 ) ⊓ ⋯ ⊓ ( ∪ σ β ) ( i r ) : x ≼ i 1 i 2 ⋯ i r , r ∈ ℕ , i 1 , . . . , i r ∈ M } ≥ ⨆ { ( ∪ σ α ) ( i 1 ) ⊓ ⋯ ⊓ ( ∪ σ α ) ( i r ) : x ≼ i 1 i 2 ⋯ i r , r ∈ ℕ , i 1 , . . . , i r ∈ M } = C ( Y ) ( x ) . Next, for X⊂FI (M) and x ∈ M we have C ( C ( X ) ) ( x ) = Fg ( ∪ C ( X ) ) ( x ) = Fg ( ∪ Fg ( ∪ X ) ) ( x ) = Fg ( Fg ( ∪ X ) ) ( x ) = ⨆ { ( Fg ( ∪ X ) ( i 1 ) ⊓ ⋯ ⊓ ( Fg ( ∪ X ) ( i r ) : x ≼ i 1 ⋯ i r , r ∈ ℕ , i 1 , … , i r ∈ M } ≼ ⨆ { Fg ( ∪ X ) ( x ) } = Fg ( ∪ X ) ( x ) = C ( X ) ( x ) Obviously, C (X) sC (C (X)). Thus C is a closure operator.

The second part follows from the fact that for any nonempty set A of fuzzy A-ideals we have ⋃A = ⋃ {∪ X : X∈A isfinite}. Indeed, for any x ∈ M, ( ⋃ A ) ( x ) = ⨆ { ( ∪ X ) ( x ) : X∈A is finite } and so for any i₁, …, i₂ ∈ M such that x ≼ i₁  ⋯ i _r , ( ∪ A ) ( i 1 ) ⊓ ⋯ ⊓ ( ∪ A ) ( i r ) = ⨆ ( ∪ X ) ( i 1 ) ⊓ ⋯ ⊓ ⨆ ( ∪ X ) ( i r ) = ⨆ { ( ∪ X ) ( i 1 ) ⊓ ⋯ ⊓ ( ∪ X ) ( i r ) } . Hence Fg (∪ A) (x) = ⋃ {Fg (∪ X) : X∈A isfinite}. □

From Theorem 3.6 and [21 Theorem I.5.5] it follows that

Corollary 3.7. FI (M) is an algebraic lattice in which the compact elements are precisely the fuzzy A-ideals Fg (A), where A is a finite set of fuzzy A-ideals of M.

Corollary 3.8. Let A be the set of all compact elements of FI (M). Then there is an algebraic closure operator C on A such that FI (M) is isomorphic to the lattice of closed subsets of A.

Proof. Assume that C : 2 ^A ra2 ^A is defined by C (X) = {σ ∈ A : σs ⨆ X}. Since FI (M) is algebraic, C is well-defined. The remain follows from [21 Theorem I.5.8].□

Theorem 3.9. The structure FI (M) = (FI (M) ; ⊓ , ⊔), where for all σ, ν ∈ FI (M),

σ ⊓ ν = σ ∩ ν σ ⊔ ν = [ σ ∪ ν ] is a complete Brouwerian lattice.

Proof. We shall prove that σ ⊓ (⨆ ν _α) ≼ ⨆ (σ ⊓ ν _α). The converse inequality always holds. Let x ∈ M and ɛ > 0 be arbitrary. Then x ≼ a₁ a₂  ⋯ a _n and ( ⨆ ν α ) ( x ) < ɛ + ( ∪ ν α ) ( a 1 ) ⊓ ⋯ ⊓ ( ∪ ν α ) ( a n ) for some n ∈ ℕ and a₁, a₂, …, a _n  ∈ M. By the definition of ∪ν _α we get (∪ ν _α) (a _i ) < ν _{α i}  (a _i ) +  ɛ, for at least one α _i . Obviously, we can assume that

( ∪ ν α ) ( a 1 ) < ν α 1 ( a 1 ) + ɛ ( ∪ ν α ) ( a 2 ) < ν α 2 ( a 2 ) + ɛ ⋮ ( ∪ ν α ) ( a n ) < ν α n ( a n ) + ɛ .

Hence, ( ∪ ν α ) ( x ) < n ɛ + ν α 1 ( a 1 ) ⊓ ⋯ ⊓ ν α n ( a n ) and so σ ⊓ ( ∪ ν α ) ( x ) < n ɛ + ( σ ( x ) ⊓ ν α 1 ( a 1 ) ) ⊓ ⋯ ⊓ ( σ ( x ) ⊓ ν α n ( a n ) ) . Now, let b n ★ = a 1 a 2 ⋯ a n - 1 x ★ , b n - 1 ★ = a 1 a 2 ⋯ a n - 2 b n x ★ , ⋮ b 2 ★ = a 1 b n b n - 1 ⋯ b 3 x ★ , b 1 ★ = b n b n - 1 ⋯ b 2 x ★ . We observe that b 1 b 2 ⋯ b n x ★ = b 1 b 1 ★ = 0 ★ whence x ≼ b 1 b 2 ⋯ b n .(3.10) On the other hand, b n ★ a n = a 1 a 2 ⋯ a n x ★ = 0 ★ , whence b _n  ≼ a _n . Also, b 1 ★ a 1 = a 1 b n ⋯ b 3 b 2 x ★ = b 2 ★ b 2 = 0 ★ and so b₁ ≼ a₁. For i = 2, …, n - 1 we have b i ★ a i = a 1 ⋯ a i b n b n - 1 ⋯ b i + 1 x ★ = 0 ★ . Thus b _i  ≼ a _i , for each i ∈ {1, 2, …, n}, whence ν _{α i}  (a _i ) ≤ ν _{α i}  (b _i ). Obviously, b i ★ x = 0 ★ , whence b _i  ≼ x and so σ (x) ≤ σ (b _i ), for all i ∈ {1, 2, …, n}. Thus σ (x) ⊓ ν _{α i}  (a _i ) ≤ (σ ⊓ ν _{α i} ) (b _i ) and so σ ⊓ ( ∪ ν α ) ( x ) < n ɛ + ( σ ⊓ ν α 1 ) ( b 1 ) ⊓ ⋯ ⊓ ( σ ⊓ ν α n ) ( b n ) . Obviously, σ ⊓ ν α i ( b i ) ≤ [ ∪ ( σ ⊓ ν α ) ] ( b i ) , ∀ i ∈ { 1 , 2 , … , r } . Thus by (3.10), σ ⊓ ( ∪ ν α ) ( x ) < n ɛ + ⊓ i = 1 n [ ∪ ( σ ⊓ ν α ) ] ( b i ) ≤ n ɛ + [ ∪ ( σ ⊓ ν α ) ] ( x ) Since ɛ is arbitrary, we have σ ⊓ ( ⨆ ν α ) ( x ) ≤ ⨆ ( σ ⊓ ν α ) ( x ) . □ By Theorem 3.9 and [6 Theorem 7.10] it follows that

Corollary 3.10. FI (M) = (FI (M) ; ⊓ , ⊔) is a Heyting lattice.

By Corollary 3.1, for any ν ∈ FI (M), the mapping α ↦ α ⊓ ν on FI (M) is residuated and so by [6 Theorem 1.3] sup {ν ∈ FI (M) : α ⊓ νsβ} = α→β exists, for any α, β ∈ FI (M). So, (FI (M) , ⊔ , ⊓ , →, zero, unit) is a Heyting algebra.

Furthermore, for ν ∈ FI (M), ν^★ = ν→0 is the pseudocomplement of ν. So, we have

Corollary 3.11. (FI (M) , ⊔ , ⊓) is a pseudocomplemented lattice.

Let S (FI (M)) = {ν^★ : ν ∈ FI (M)} and define α ⊔ β = (α^★ ⊓ β^★) ^★, for any α, β ∈ FI (M). Then by [6 Theorem 7.2] we have

Theorem 3.12. (S (FI (M)) , ⊓ , ⊔ , ^★, zero, unit) is a Boolean algebra.

As an applied example of Theorems 3.9 and 3.12 and Corollary 3.11 we get that FI (M) is a distributive lattice and so by [16, Theorem 101], FI (M) does not contain any pentagon or diamond as a sublattice. This means that there are not three fuzzy A-ideals (or three non-comparable fuzzy A-ideals) μ, ν and ρ with μ ⊂ ν and μ ⊓ ρ = ν ⊓ ρ and μ ⊔ ρ = ν ⊔ ρ. For convenience, we consider the crisp case. If I, J and K are ideals of M with I ⊂ J, then there exists an ideal K such that I ∩ K = J ∩ K. In other words, the elimination property for the intersection operation does not hold. Similarly, for the join operation we get a same result. Furthermore, for any ideal I the largest ideal K such that I ⊓ K = zero exists, which is called the pseudocomplement of I. Most important result is that these pseudocomplements form a Boolean lattice (or Boolean algebra).

An element ϱ ∈ FR (M) is called a fuzzy equivalence relation if for any x, y ∈ M, (i) ϱ (x, x) = ⨆ {ϱ (y, z) : y, z ∈ M}, (ii) ϱ (x, y) = ϱ (y, x), and ϱ (x, y) ≥ ⨆ _z∈M {ϱ (x, z) ⊓ ϱ (z, y)}. Let Fe (M) denotes the set of all fuzzy equivalence relations in M. The intersection, composition and the inclusion relation on [0, 1] ^M×M are defined as

ϱsσ ⇔ ϱ (x, y) ≤ σ (x, y),

Definition 3.13. Let ϑ ∈ Fe (M). ϑ is called a fuzzy congruence if for any x, y, u, v ∈ M,

ϑ (x^★, y^★) ≥ ϑ (x, y),

Example 3.14. Consider the mv-module M defined in Example 3.1 and define the fuzzy relation ϱ by ϱ (x, x) = t for all x ∈ M and ϱ (x, y) = s, for all x ≠ y, where 0 ≤ s < t ≤ 1. It is easy to check that ϱ is a fuzzy congruence in M.

We say that M is fuzzy congruence-permutable if ϱccσ = σccϱ, for all ϱ, σ ∈ Fc (M). It is easy to verify that for ϱ, σ ∈ FR (M) with the same tip t, ϱ ∩ σ and ϱccσ have the same tip t.

Let con (M), FR (M) (Fe (M), Fc (M)) denote the set of all congruences and fuzzy (equivalence, congruence) relations in M, respectively.

Obviously, by (i) and the definition of the operation ‘ ’, the condition (ii) of Definition 3.13 can be replaced by ϑ ( x y , u v ) ≥ ϑ ( x , u ) ⊓ ϑ ( y , v ) .(11)

By a routine verification we find that for ϑ ∈ FR (M), ϑ ∈ Fc (M) if and only if ϑ _t  ∈ con (M), for all t ∈ [0, 1], with ϑ _t ≠ ∅.

Let ϑ ∈ FR (M). For x ∈ M, ϑ _x  ∈ [0, 1] ^M is defined as ϑ _x  (y) = ϑ (y, x), for all y ∈ M. Now, we have the following lemma.

Lemma 3.15. Let ϑ ∈ Fc (M). Then

ϑ _x  = ϑ _y if and only if ϑ ( x , y ) = ⨆ ( u , v ) ∈ M 2 ϑ ( u , v ) . Particularly, ϑ (0, 0) = ⨆ _(u,v)∈M² ϑ (u, v).

Proof. (i) Assume that ϑ _x  = ϑ _y , for x, y ∈ M. Then ϑ ( x , y ) = ϑ y ( x ) = ϑ x ( x ) = ϑ ( x , x ) = ⨆ ( u , v ) ∈ M 2 ϑ ( u , v ) . Conversely, for w ∈ M, we have ϑ x ( a ) = ϑ ( a , x ) = ϑ ( x , a ) ≥ ϑ ( x , y ) ⊓ ϑ ( y , a ) = ( ⨆ ( u , v ) ∈ M 2 ϑ ( u , v ) ) ⊓ ϑ ( y , a ) = ϑ ( y , a ) = ϑ y ( a ) . Similarly, it is proved that for all u ∈ M, ϑ _y  (u) ≥ ϑ _x  (u). Hence, ϑ _x  = ϑ _y .

(ii) Assume that ϑ _t ≠ ∅, for t ∈ [0, 1] and x ∈ 0/ϑ _t . Then (x, 0) ∈ ϑ _t ; i.e., ϑ₀ (x) = ϑ (x, 0) ≥ t, whence x ∈ (ϑ₀)  _t . Then 0/ϑ _t s (ϑ₀)  _t . Conversely, if x ∈ (ϑ₀)  _t , then ϑ (x, 0) = ϑ₀ (x) ≥ t, whence x ∈ 0/ϑ _t . Hence, (ϑ₀)  _t s0/ϑ _t , completes the proof.□

Theorem 3.16. Let ϑ ∈ Fc (M). Then M/ϑ = {ϑ _x  : x ∈ M} together with the following operations forms an mv-module.

ϑ x ϑ y = ϑ x y , ( ϑ x ) ★ = ϑ x ★ and ∀ α ∈ A , α ϑ x = ϑ α x .

Proof. It is enough to prove that the mentioned operations are well-defined. Assume that ϑ _x  = ϑ _z and ϑ _y  = ϑ _w , for ϑ _x , ϑ _y , ϑ _z , ϑ _w  ∈ M/ϑ. By Lemma 3.15 we have ϑ (x, z) = ⨆ _(u,v)∈^M2 ϑ (u, v) = ϑ (y, w). This implies that (x, z) , (y, w) ∈ ϑ _t , where t = ⨆ _(u,v)∈^M2 ϑ (u, v), whence (x y, z w) ∈ ϑ _t . Hence, ⨆ { ϑ ( u , v ) : ( u , v ) ∈ M 2 } ≥ ϑ ( x y , z w ) ≥ t = ⨆ { ϑ ( u , v ) : ( u , v ) ∈ M 2 } . Then ϑ (x y, z w) = ⨆ _(u,v)∈^M2 ϑ (u, v) and so ϑ x ϑ y = ϑ x y = ϑ z w = ϑ z ϑ w . Now, if ϑ _x  = ϑ _y , then ϑ ( x ★ , y ★ ) ≥ ϑ ( x , y ) = ⨆ { ϑ ( u , v ) : ( u , v ) ∈ M 2 } , which implies that ϑ (x^★, y^★) = ⨆ _(u,v)∈^M2 ϑ (u, v). Thus, (ϑ _x ) ^★ = ϑ _{x ^★}  = ϑ _{y ^★}  = (ϑ _y ) ^★. The other operation is obviously well-defined.□ We call the mv-module M/ϑ, in Theorem 3.16, the fuzzy quotient mv-module of M.

Proposition 3.17. If ϑ ∈ Fc (M), then ϑ₀ ∈ fai (M).

Proof. Assume that ϑ ∈ Fc (M). Then, for every a, b ∈ M, ϑ 0 ( a b ) = ϑ ( a b , 0 ) ≥ ϑ ( a , 0 ) ⊓ ϑ ( b , 0 ) = ϑ 0 ( a ) ⊓ ϑ 0 ( b ) . Now, let a ≼ b, for a, b ∈ M. Then b^★ a = 0 and since 0^★ a = a we get ϑ 0 ( a ) = ϑ ( a , 0 ) = ϑ ( 0 ★ a , b ★ a ) ≥ ϑ ( 0 ★ , b ★ ) ⊓ ϑ ( a , a ) ≥ ϑ 0 ( b ) . Finally, for a ∈ M and α ∈ A we have ϑ 0 ( α a ) = ϑ ( α a , 0 ) = ϑ ( α a , α 0 ) ≥ ϑ ( a , 0 ) = ϑ 0 ( a ) . Therefore, ϑ₀ ∈ fai (M).□

Example 3.18. Consider the mv-module M defined in Example 3.1. We define the σ ∈ FR (M) by σ (x, x) = t₃, for all x ∈ M, σ (∅ , {2}) = σ ({2} , ∅) = t₁, and σ (x, y) = t₂, for the other elements of M, where 0 ≤ t₁ < t₂ < t₃ ≤ 1. It is obvious that σ_∅ = σ is a fuzzy A-ideal of M, while σ ∉ Fc (M) because σ ({1} ^★, {1, 2} ^★) = σ ({2} , ∅) = t₁notget₂ = σ ({1} , {1, 2}).

This shows that the converse of Proposition 3.2 is not true in general.

For fuzzy relation ϑ in M, let M ϑ = { x ∈ M : ϑ ( x ) = ϑ ( 0 ) } .

Theorem 3.19. Let ϑ ∈ Fc (M) and f : M→N be a homomorphism of mv-modules with _{M ϑ0}sKerf. Then the mapping f ˆ : M / ϑ →N with ϑ _x  ↦ f (x) is the unique homomorphism with Im f ˆ = Imf and Ker f ˆ = Kerf / ϑ . Furthermore, f ˆ is an isomorphism if and only if f is an epimorphism and Kerf = _Mϑ0.

Proof. Assume that f ˆ is defined as f ˆ ( ϑ x ) = f ( x ) . Assume that ϑ _x  = ϑ _y , for ϑ _x , ϑ _y  ∈ M/ϑ. Then ϑ (x, y) = t, where t = ⨆ _(u,v)∈^M2 ϑ (u, v). Hence ϑ 0 ( x y ★ ) = ϑ ( x y ★ , 0 ) = ϑ ( x y ★ , y y ★ ) ≥ ϑ ( x , y ) ∧ ϑ ( y ★ , y ★ ) = t = ϑ ( 0 , 0 ) = ϑ 0 ( 0 ) , by Lemma 3.15(i), whence x y^★ ∈ _Mϑ0⊂Kerf and so f (x)  f (y) ^★ = f (x y^★) =0; i.e., f (x) ≼ f (y). Similarly, it is proved that f (y) ≼ f (x). Thus, f ˆ is well-defined. It is clear that Im f ˆ = Imf . Now, ϑ x ∈ Ker f ˆ if and only if f ( x ) = f ˆ ( ϑ x ) = 0 ; i.e., if and only if x ∈ Kerf. This means that Ker f ˆ = Kerf / ϑ .

We observe that if f ˆ is an epimorphism, f is also an epimorphism, and conversely. Also, f ˆ is one-to-one if and only if Kerf / ϑ = Ker f ˆ = { ϑ 0 } and this is true if and only if Kerf = _Mϑ0.□

Theorem 3.20. Let ϑ ∈ Fc (M) and f : M→N an epimorphism of mv-modules such that Kerf = _Mϑ0. Then M/ϑ ≃ N.

Proof. From Theorem 3.19, taking N = Imf, the proof is obvious.□

Theorem 3.21. Let ϑ ∈ Fc (M). Then M/ϑ ≃ M/_{M ϑ0}.

Proof. We first observe that _{M ϑ0} is an A-ideal of M and under the congruence ∼ _{M ϑ0} , M/_{M ϑ0} is an mv-module. It is clear that φ : M→M/_{M ϑ0} with x ↦ x/_{M ϑ0} is an epimorphism of mv-modules with the kernel _{M ϑ0}. So, by Theorem 3.20, the proof is complete.□

Lemma 3.22. Let ϑ, S ∈ Fc (M). Then ϑ/S ∈ Fc (M/S), where ϑ/S (_Sx, _Sy) = ϑ (x, y).

Proof. Straightforward.□

Theorem 3.23. Let ϑ, S ∈ Fc (M) with the same tip such that S⊂ϑ. Then (M/S)/(ϑ/S) ≃ M/ϑ.

Proof. Let φ : M/S→M/ϑ be defined by _Sx ↦ ϑ _x . We first show that φ is well-defined. Let _Sx = _Sy. Then ϑ ( x , y ) ≥ ⋃ ( x , y ) = ⨆ { ⋃ ( u , v ) : ( u , v ) ∈ M 2 } = ⨆ { ϑ ( u , v ) : ( u , v ) ∈ M 2 } , whence ϑ (x, y) = ⨆ {ϑ (u, v) : (u, v) ∈ ^M2}. Now, by Lemma 3.15 we have ϑ _x  = ϑ _y , proving φ is well-defined. Clearly, φ is an epimorphism. Now, Ker φ = { ⋃ x ∈ M / ⋃ : ϑ x = φ ( ⋃ x ) = ϑ 0 } = { ⋃ x ∈ M / ⋃ : ϑ ( x , 0 ) = ϑ ( 0 , 0 ) } = { ⋃ x ∈ M / ⋃ : ϑ / ⋃ ( ⋃ x , ⋃ 0 ) = ϑ / ⋃ ( ⋃ 0 , ⋃ 0 ) } = { ⋃ x ∈ M / ⋃ : ( ϑ / ⋃ ) ⋃ 0 ( ⋃ x ) = ( ϑ / ⋃ ) ⋃ 0 ( ⋃ 0 ) } = M ( ϑ / ⋃ ) ⋃ 0 . Thus, by Theorem 3.20, we get (M/S)/(ϑ/S) ≃ M/ϑ.

□

In Theorem 3.16, we proved that if ϑ ∈ Fc (M), M/ϑ together with the operations induced from those of M forms an mv-module without giving any information about the existence of ϑ. Here, we show that any fuzzy A-ideal induces such a fuzzy congruence.

For x, y ∈ M define m (x, y) = (x y^★) y ( x^★). It is routine to verify that

(m1)  m (x, 0) = x.

(m2)  m (x, y) ≼ m (x, z) m (z, y).

(m3)  m (x u, y v) ≤ m (x, y) m (u, v).

(m4)  m (ax, ay) ≼ am (x, y).

Definition 3.24. For fuzzy subset σ of M we define ϑ _σ ∈ FR (M) by ϑ σ ( x , y ) = σ ( m ( x , y ) ) , ∀ x , y ∈ M .

Theorem 3.25. ϑ _σ ∈ Fc (M).

Proof. We first prove that ϑ _σ ∈ Fe (M). Let x ∈ M. From σ (m (x, x)) = σ (0) ≥ σ (u), for all u ∈ M, it follows that σ (m (x, x)) ≥ σ (m (y, z)), for all y, z ∈ M. Hence, ϑ σ ( x , x ) = σ ( m ( x , x ) ) = ⨆ { σ ( m ( y , z ) ) : ( y , z ) ∈ M 2 } = ⨆ { ϑ σ ( y , z ) : ( y , z ) ∈ M 2 } , i.e., ϑ _σ is fuzzy reflexive. Obviously, ϑ _σ is fuzzy symmetric. Now, let x, y ∈ M. By (m2) it follows that ϑ σ ( x , y ) = σ ( m ( x , y ) ) ≥ σ ( m ( x , z ) m ( z , y ) ) ≥ σ ( m ( x , z ) ) ⊓ σ ( m ( z , y ) ) = ϑ σ ( x , z ) ⊓ ϑ σ ( z , y ) and so ϑ _σ (x, y) ≥ ⨆ _z∈M ϑ _σ (x, z) ⊓ ϑ _σ (z, y), proving that ϑ _σ is fuzzy transitive. Therefore, ϑ _σ ∈ Fe (M). That ϑ _σ is a fuzzy congruence is a direct consequence of (m3) and (m4).□

Theorem 3.26. Let σ and ν be fuzzy A-ideals of M. Then

ϑ σ ∩ ν = ϑ σ ∩ ϑ ν ϑ σ ⊔ ν = ϑ σ ⊔ ϑ ν

Proof. We prove that ϑ _σ⊔ν is the least fuzzy congruence on M which contains ϑ _σ and ϑ _ν. From Theorem 3.25, we know that ϑ _σ, ϑ _ν and ϑ _σ⊔ν are fuzzy congruence on M. Now, for x, y ∈ M, ϑ σ ( x , y ) = σ ( m ( x , y ) ) ≤ σ ⊔ ν ( m ( x , y ) ) = ϑ σ ⊔ ν ( x , y ) . Similarly, ϑ _ν (x, y) ≤ ϑ _σ⊔ν (x, y). Hence ϑ _σ ⊔ ϑ _νsϑ _σ⊔ν. Now, let SFc (M) with ϑ _σ⊂S and ϑ _ν⊂S. Then σ (m (x, y)) = ϑ _σ (x, y) ≤ S (x, y) and ν (m (x, y)) = ϑ _ν (x, y) ≤ S (x, y). Also, ϑ σ ⊔ ν ( x , y ) = σ ⊔ ν ( m ( x , y ) ) = ⨆ { σ ∪ ν ( i 1 ) ⊓ ⋯ ⊓ σ ∪ ν ( i n ) : m ( x , y ) ≼ i 1 i 2 ⋯ i n } = ⨆ { ( σ ( i 1 ) ⊓ ⋯ ⊓ σ ( i n ) ) ∪ ( ν ( i 1 ) ⊓ ⋯ ⊓ ν ( i n ) ) : m ( x , y ) ≼ i 1 i 2 ⋯ i n } ≤ ⨆ { σ ( i 1 i 2 ⋯ i n ) ∪ ν ( i 1 i 2 ⋯ i n ) : m ( x , y ) ≼ i 1 i 2 ⋯ i n } ≤ ⊔ { σ ( m ( x , y ) ) ∪ ν ( m ( x , y ) ) : m ( x , y ) ≼ i 1 i 2 ⋯ i n } ≤ ⋃ ( x , y ) proving ϑ _σ⊔ν = ϑ _σ ⊔ ϑ _ν. Obviously, ϑ _σ ∩ ϑ _ν = ϑ _σ∩ν.□

Remark 3.27. From [2 Theorem 3.9], we know that if M has tip property and is fuzzy congruence-permutable, then ϑ _σ ⊔ ϑ _ν = ϑ _σ ∘ ϑ _ν, where ∘ is the composition operation of fuzzy relations.

Theorem 3.28. Assume that M has tip property and is fuzzy congruence-permutable. Then the lattice FI (M) can be embedded into the lattice FC (M) = (FC ; ∩ , cc).

Proof. We define the mapping φ : FI→FC by σ ↦ ϑ _σ. By Theorem 3.26, φ preserves ⊓ and ⊔. Now, define ψ : FC→FI by ϑ ↦ ϑ₀. By Proposition 3.2, ψ is well-defined and also ψ φ ( σ ) ( x ) = ψ ( ϑ σ ) ( x ) = ( ϑ σ ) 0 ( x ) = ϑ σ ( x , 0 ) = σ ( m ( x , 0 ) ) = σ ( x ) , whence φ is an embedding.□

By Theorems 3.9 and 3.28 it follows that

Corollary 3.29. If M has tip property and is fuzzy congruence-permutable, then FI (M) is a Heyting sublattice of FC (M).

Open Problem 3.30. Under what conditions FI and FC are isomorphic?