Let H = { 0 , a , b , 1 } be a lattice with 0 < a < 1 , 0 < b < 1 , and a and b incomparable. Define “ → ” as x → y = { 1 , x ≤ y y , otherwise . Then ( H , ∨ , ∧ , → , 0 , 1 ) is a Heyting algebra. Let H be a nonempty set. A fuzzy subset μ of H is a function μ: H → [0, 1].

Let H = { 0 , a , b , 1 } as in Example 2.2. Define μ : H → [ 0 , 1 ] by : μ ( 0 ) = 1.0 , μ ( a ) = 0.7 , μ ( b ) = 0.5 , μ ( 1 ) = 0.3 . Then μ is a fuzzy subset of H.

A fuzzy subset μ of a Heyting algebra H is called: 1.

Fuzzy multiplicatively closed if μ ( a ) ∧ μ ( b ) ≤ μ ( a ∧ b ) for all a , b ∈ H .

Let H be as in Example 2.2. Define ν : H → [ 0 , 1 ] by ν ( 0 ) = 0.2 , ν ( a ) = 0.6 , ν ( b ) = 0.6 , ν ( 1 ) = 1.0 . Then ν is fuzzy multiplicatively closed since for any y ∈ H , min ( ν ( x ) , ν ( y ) ) ≤ ν ( x ∧ y ) .

A fuzzy subset μ of H is called a fuzzy ideal if: 1.

μ ( 0 ) = 1

Let H = {0, a, b, 1} as before. Define: μ ( 0 ) = 1.0 , μ ( a ) = 0.8 , μ ( b ) = 0.8 , μ ( 1 ) = 0.9 . Then μ is a fuzzy ideal.

A fuzzy subset ν of H is called a fuzzy filter if: 1.

ν ( 1 ) = 1

Define ν : H → [ 0 , 1 ] by: ν ( 0 ) = 0.3 , ν ( a ) = 0.7 , ν ( b ) = 0.7 , ν ( 1 ) = 1.0 .

Then ν is a fuzzy filter.

A fuzzy ideal μ is prime if for all a , b ∈ H , μ ( a ∧ b ) ≤ max ( μ ( a ) , μ ( b ) ) .

A fuzzy filter ν is prime if for all a , b ∈ H , ν ( a ∨ b ) ≤ max ( ν ( a ) , ν ( b ) ) . The foundational work on congruence relations in Heyting algebras by Assaye et al. ² established several key results that our fuzzy extensions build upon:

(Assaye et al., 2019). For any prime ideal P and a filter F of a Heyting algebra H, there exists an order-preserving map between the set of all prime ideals of H / ψ S and the set of all prime ideals of H disjoint from S, where ψ S is a special congruence relation induced by an implicatively closed subset S. This classical result provides the template for our fuzzy extension in Theorem 4.30.

Let H be a Heyting algebra. A fuzzy relation θ : H × H → [ 0 , 1 ] is called a fuzzy congruence relation if for all a , b , c , d ∈ H : 1.

θ ( a , a ) = 1 (Fuzzy reflexivity)

Let H = { 0 , a , b , 1 } as before. Define θ : H × H → [ 0 , 1 ] by: ( x , y ) = { 1 if x = y 0.7 if { x , y } = { 0 , a } or { 0 , b } 0.5 if { x , y } = { a , b } 0.3 , otherwise

We can verify this satisfies the fuzzy congruence conditions for appropriate Heyting algebra structures. Extending Definition 3.1 with insights from, ¹⁴ we introduce:

(Strong Fuzzy Congruence). A fuzzy relation θ : H × H → [ 0 , 1 ] is a strong fuzzy congruence if it satisfies all conditions of Definition 3.1 plus the additional condition: θ ( a , b ) = θ ( a → c , b → c ) ∧ θ ( c → a , c → b ) , ∀ a , b , c ∈ H .

Let H = { 0 , a , b , 1 } with ˜ S defined by: ˜ S ( 0 ) = 0.2 , ˜ S ( a ) = 0.7 , ˜ S ( b ) = 0.7 , ˜ S ( 1 ) = 1.0

Then ˜ S is fuzzy implicatively closed. One can compute Ψ ˜ S ( a , b ) for some pairs.

If ˜ S is a fuzzy implicatively closed subset of H , then Ψ ˜ S is a fuzzy congruence relation on H.

Let ˜S be a fuzzy implicatively closed subset of H. We verify each condition: For fuzzy reflexivity: For any a ∈ H , Ψ ˜ S ( a , a ) = in f s ∈ H ( ˜ S ( s ) → θ s ( a , a ) ) , where θ s ( a , a ) = min ( ˜ S ( a → s ) , ˜ S ( a → s ) ) = ˜ S ( a → s ) . Since ˜ S ( s ) → ˜ S ( a → s ) = 1 for all s (by definition of → in [0,1]), the infimum is reflexive. Fuzzy symmetry is immediate since θ s ( a , b ) = θ s ( b , a ) .

For fuzzy transitivity, we need to show: Ψ ˜ S ( a , b ) ∧ Ψ ˜ S ( b , c ) ≤ Ψ ˜ S ( a , c ) .

For any s ∈ H : and Thus: ˜ S ( s ) → θ s ( a , b ) ≥ Ψ ˜ S ( a , b ) and ˜ S ( s ) → θ s ( b , c ) ≥ Ψ ˜ S ( b , c ) ⇒ ˜ S ( s ) → ( θ s ( a , b ) ∧ θ s ( b , c ) ) ≥ Ψ ˜ S ( a , b ) ∧ Ψ ˜ S ( b , c )

Since ˜ S is implicatively closed, we have: ˜ S ( a → s ) ∧ ˜ S ( b → s ) ≤ ˜ S ( ( a → s ) ∧ ( b → s ) ) ≤ ˜ S ( a → c ) when b → s ≤ a → c . This establishes the transitivity condition.

For preservation of ∧, consider for any s: θ s ( a ∧ c , b ∧ d ) = min ( ˜ S ( ( a ∧ c ) → s ) , ˜ S ( ( b ∧ d ) → s ) ) .

Using Heyting algebra identity ( a ∧ c ) → s = a → ( c → s ) and the fact that ˜S preserves implications, we get: min ( ˜ S ( a → s ) , ˜ S ( c → s ) ) ∧ min ( ˜ S ( b → s ) , ˜ S ( d → s ) ) ≤ θ s ( a ∧ c , b ∧ d ) .

This implies the condition. Preservation of ∨ and → follows from similar arguments using distributive laws and properties of → .

Thus, Ψ ˜ S is a fuzzy congruence relation on H.

Let θ be a fuzzy congruence on H. The fuzzy kernel of θ is: Ker ( θ ) ( a ) = θ ( a , 0 ) , ∀ a ∈ H . Example 4.5. For the θ in Example 3.2: Ker ( θ ) ( 0 ) = θ ( 0 , 0 ) = 1 , Ker ( θ ) ( a ) = θ ( a , 0 ) = 0.7 , Ker ( θ ) ( b ) = 0.7 , Ker ( θ ) ( 1 ) = 0.3 .

If θ is a fuzzy congruence on H, then Ker ( θ ) is a fuzzy ideal of H.

Let θ be a fuzzy congruence on H. First, Ker ( θ ) ( 0 ) = θ ( 0 , 0 ) = 1 by reflexivity.

Second, for a , b ∈ H : Ker ( θ ) ( a ) ∧ Ker ( θ ) ( b ) = θ ( a , 0 ) ∧ θ ( b , 0 ) . By the preservation of ∨ (condition 5 in Definition 3.1) θ ( a , 0 ) ∧ θ ( b , 0 ) ≤ θ ( a ∨ b , 0 ∨ 0 ) = θ ( a ∨ b , 0 ) = Ker ( θ ) ( a ∨ b ) . Third, for any b ∈ H : Ker ( θ ) ( a ) = θ ( a , 0 ) ≤ θ ( a ∧ b , 0 ∧ b ) = θ ( a ∧ b , 0 ) = Ker ( θ ) ( a ∧ b ) using preservation of ∧ and the fact that 0 ∧ b = 0 .

Thus , Ker ( θ ) satisfies all conditions of a fuzzy ideal.

Let μ be a fuzzy ideal of H. Define a fuzzy relation θ μ on H by:

θ μ ( a , b ) = μ ( a ↔ b ) , where a ↔ b = ( a → b ) ∧ ( b → a ) . Then, θ μ is a fuzzy congruence on H and Ker ( θμ ) = μ .

Let μ be a fuzzy ideal. Define θ μ ( a , b ) = μ ( a ↔ b ) where a ↔ b = ( a → b ) ∧ ( b → a ) .

First, we show θ μ is a fuzzy congruence: Reflexivity: θ μ ( a , a ) = μ ( a ↔ a ) = μ ( 1 ) = 1 since μ ( 1 ) = μ ( 0 → 0 ) ≥ μ ( 0 ) = 1 . Symmetry: θ μ ( a , b ) = μ ( a ↔ b ) = μ ( b ↔ a ) = θ μ ( b , a ) .

Transitivity: We need μ ( a ↔ b ) ∧ μ ( b ↔ c ) ≤ μ ( a ↔ c ) .

In any Heyting algebra, ( a ↔ b ) ∧ ( b ↔ c ) ≤ ( a ↔ c ) .

Since μ is a fuzzy ideal: μ ( a ↔ b ) ∧ μ ( b ↔ c ) ≤ μ ( ( a ↔ b ) ∧ ( b ↔ c ) ) ≤ μ ( a ↔ c )

Preservation of ∧ : θ μ ( a , c ) ∧ θ μ ( b , d ) = μ ( a ↔ c ) ∧ μ ( b ↔ d ) ≤ μ ( ( a ↔ c ) ∧ ( b ↔ d ) ) .

In Heyting algebras, ( a ↔ c ) ∧ ( b ↔ d ) ≤ ( a ∧ b ) ↔ ( c ∧ d ) , so : ≤ μ ( ( a ∧ b ) ↔ ( c ∧ d ) ) = θ μ ( a ∧ b , c ∧ d ) .

Preservation of ∨ and → follows from similar arguments using Heyting algebra identities.

For the kernel property: Ker ( θμ ) ( a ) = θ μ ( a , 0 ) = μ ( a ↔ 0 ) = μ ( a ∧ ( 0 → a ) ) = μ ( a ) .

Since a ↔ 0 = a in Heyting algebras. Thus θ μ is a fuzzy congruence and Ker ( θ μ ) = μ .

Let H = { 0 , a , b , 1 } with μ as in Example 2.6: μ ( 0 ) = 1.0 , μ ( a ) = 0.8 , μ ( b ) = 0.8 , μ ( 1 ) = 0.9 .

Then, θ μ ( a , b ) = μ ( a ↔ b ) . Compute a ↔ b : a → b = b b → a = a a ↔ b = a ∧ b = 0

So θ μ ( a , b ) = μ ( 0 ) = 1 .

Let μ be a fuzzy ideal. The fuzzy congruence generated by μ is: θ μ ∗ ( a , b ) = sup { μ ( c ) : c ∈ H , a ↔ b ≥ c } .

With μ as above and a , b with a ↔ b = 0 , then: θ μ ∗ ( a , b ) = sup { μ ( c ) : 0 ≥ c } = μ ( 0 ) = 1

θ μ ∗ is the smallest fuzzy congruence on H whose kernel contains μ.

Let θ be a fuzzy congruence on H . The fuzzy cokernel is: Coker ( θ ) ( a ) = θ ( a , 1 ) .

For θ from Example 3.2: Coker ( θ ) ( 0 ) = θ ( 0 , 1 ) = 0.3 , Coker ( θ ) ( a ) = 0.3 , Coker ( θ ) ( b ) = 0.3 , Coker ( θ ) ( 1 ) = 1 .

If θ is a fuzzy congruence, then Coker ( θ ) is a fuzzy filter of H.

Let θ be a fuzzy congruence. First, Coker ( θ ) ( 1 ) = θ ( 1 , 1 ) = 1 .

Second, for b ∈ H : Coker ( θ ) ( a ) ∧ Coker ( θ ) ( b ) = θ ( a , 1 ) ∧ θ ( b , 1 ) ≤ θ ( a ∧ b , 1 ∧ 1 ) = θ ( a ∧ b , 1 ) = Coker ( θ ) ( a ∧ b ) .

Third, for any b ∈ H : Coker ( θ ) ( a ) = θ ( a , 1 ) ≤ θ ( a ∨ b , 1 ∨ b ) = θ ( a ∨ b , 1 ) = Coker ( θ ) ( a ∨ b )

Thus, Coker ( θ ) satisfies all conditions of a fuzzy filter.

(Characterization via Cokernel Filter). Let ν be a fuzzy filter of H .

Define: θ ν ( a , b ) = ν ( ( a → b ) ∧ ( b → a ) ) . Then θ ν is a fuzzy congruence and Coker ( θ ν ) = ν .

Let ν be as in Example 2.8. Then: θν ( a , b ) = ν ( ( a → b ) ∧ ( b → a ) ) = ν ( a ∧ b ) = ν ( 0 ) = 0.3 .

A fuzzy filter ν is implicative if: ν ( a → b ) ∧ ν ( a ) ≤ ν ( b ) .

Define ν : H → [ 0 , 1 ] by : ν ( 0 ) = 0.4 , ν ( a ) = 0.6 , ν ( b ) = 0.6 , ν ( 1 ) = 1

Check: ν ( a → b ) = ν ( b ) = 0.6 , ν ( a ) = 0.6 , ν ( b ) = 0.6 , so 0.6 ∧ 0.6 = 0.6 ≤ 0.6 holds.

For a fuzzy implicative filter ν , the relation: θ ν ( a , b ) = ν ( a → b ) ∧ ν ( b → a ) is a fuzzy congruence, and ν = Coker ( θ ν ) .

Let ν be a fuzzy implicative filter. Define θ ν ( a , b ) = ν ( a → b ) ∧ ν ( b → a ) .

Reflexivity: θ ν ( a , a ) = ν ( a → a ) ∧ ν ( a → a ) = ν ( 1 ) ∧ ν ( 1 ) = 1 .

Symmetry is immediate from the definition.

Transitivity requires ν ( a → b ) ∧ ν ( b → a ) ∧ ν ( b → c ) ∧ ν ( c → b ) ≤ ν ( a → c ) ∧ ν ( c → a ) from implicative filter property: ν ( a → b ) ∧ ν ( b → c ) ≤ ν ( a → c ) .

Similarly: ν ( c → b ) ∧ ν ( b → a ) ≤ ν ( c → a ) .

Thus, the inequality holds. Preservation of operations follows from Heyting algebra identities and filter properties. For the cokernel: Coker ( θν ) ( a ) = θ ν ( a , 1 ) = ν ( a → 1 ) ∧ ν ( 1 → a ) = ν ( 1 ) ∧ ν ( a ) = ν ( a ) .

Since ν ( 1 ) = 1 and ν ( a ) ≤ ν ( 1 ) .

Thus θ ν is a fuzzy congruence and ν = Coker ( θ ν ) .

With ν above: θ ν ( a , b ) = ν ( a → b ) ∧ ν ( b → a ) = ν ( b ) ∧ ν ( a ) = 0.6 ∧ 0.6 = 0.6 . Recent work by Zhao et al. ²¹ on fuzzy nuclei in residuated lattices inspires the following connection:

Let H be a Heyting algebra. There is a bijective correspondence between: 1.

Fuzzy nuclei j : H → [ 0 , 1 ] H (fuzzy closure operators preserving ∧)

(Sketch) Given a fuzzy nucleus j , define θ j ( a , b ) = min ( j ( a ) ( b ) , j ( b ) ( a ) ) . Conversely, given θ , define j θ ( a ) ( x ) = θ ( a , a ∧ x ) . The verification follows patterns established in. ²¹ Building on Hjek’s work on fuzzy logic ⁸ and recent developments in fuzzy intuitionistic logic ²² :

(Fuzzy Intuitionistic Congruence). A fuzzy congruence θ on a Heyting algebra H is intuitionistic if for all a,b,c ∈ H:

θ ( a , b ) ≤ θ ( ¬ ¬ a , ¬ ¬ b ) where ¬ x = x → 0 is the intuitionistic negation

Every fuzzy congruence θ on a Heyting algebra H induces a fuzzy congruence θ ∗ ∗ on the Boolean algebra H ∗ ∗ = { a ∗ ∗ : a ∈ H } via : θ ∗ ∗ ( a ∗ ∗ , b ∗ ∗ ) = sup { θ ( x , y ) : x ∗ ∗ = a ∗ ∗ , y ∗ ∗ = b ∗ ∗ } .

This extends the classical Glivenko theorem to the fuzzy setting and connects with recent work on fuzzy Boolean algebras. ²⁰

Given a fuzzy congruence θ on H, we define the fuzzy quotient set H / θ as the set of fuzzy equivalence classes [ a ] θ , where the membership degree of x in [ a ] θ is θ ( a , x ) .

For θ from Example 3.2, the equivalence classes are: 1.

[ 0 ] θ with memberships: θ ( 0 , 0 ) = 1.0 , θ ( 0 , a ) = 0.7 , θ ( 0 , b ) = 0.7 , θ ( 0 , 1 ) = 0.3

The fuzzy quotient set H / θ can be equipped with operations ∧ , ∨ , → such that: [ a ] θ ∧ [ b ] θ = [ a ∧ b ] θ , [ a ] θ ∨ [ b ] θ = [ a ∨ b ] θ , [ a ] θ → [ b ] θ = [ a → b ] θ .

Then H / θ forms a Heyting algebra in the fuzzy sense, called the fuzzy quotient Heyting algebra.

(Kernel Cokernel Duality). Let θ be a fuzzy congruence on H. Then: 1.

Ker ( θ ) is a fuzzy ideal.

With θ from Example 3.2 and a , b : Ker ( θ ) ( a ↔ b ) = Ker ( θ ) ( 0 ) = 1.0 , but θ ( a , b ) = 0.5 .

This shows the equality may require specific definitions of ↔ in the fuzzy context.

A fuzzy congruence θ is prime if whenever θ ( a ∧ b , 0 ) = 1 , then either θ ( a , 0 ) = 1 or θ ( b , 0 ) = 1 .

Define θ by: θ ( x , y ) = { 1 if x = y 0.8 if ( x , y ) = ( 0 , a ) or ( a , 0 ) 0.6 otherwise

For a ∧ b = 0 , θ ( a ∧ b , 0 ) = θ ( 0 , 0 ) = 1 . We need θ ( a , 0 ) = 1 or θ ( b , 0 ) = 1 , but both are 0.8 , so not prime.

(Fuzzy Prime Correspondence). There is a bijection between: 1.

Prime fuzzy congruence on H,

We establish the bijection between prime fuzzy congruence and fuzzy prime ideals. Given a prime fuzzy congruence θ , Ker ( θ ) is a fuzzy ideal by Theorem 4.6. For the prime condition: If μ ( a ∧ b ) = 1 , then θ ( a ∧ b , 0 ) = 1 .

By primness of θ , either θ ( a , 0 ) = 1 or θ ( b , 0 ) = 1 , so μ ( a ) = 1 or μ ( b ) = 1 . Conversely, given a fuzzy prime ideal μ , define θ μ ( a , b ) = μ ( a ↔ b ) . We need to show it’s prime:

If θ μ ( a ∧ b , 0 ) = 1 , then μ ( a ∧ b ) = 1 . By primeness, μ ( a ) = 1 or μ ( b ) = 1 , so θ μ ( a , 0 ) = 1 or θ μ ( b , 0 ) = 1 . For the correspondence with fuzzy prime filters:

Given θ , Coker ( θ ) is a fuzzy filter by Theorem 4.14. For primeness:

If ν ( a ∨ b ) = 1 , then θ ( a ∨ b , 1 ) = 1 . By congruence properties and primeness, either θ ( a , 1 ) = 1 or θ ( b , 1 ) = 1 . Conversely, given a fuzzy prime filter ν , define θ ν as in Theorem 4.19 and verify primeness. The maps are inverses: θ → Ker ( θ ) → θ Ker ( θ ) equals θ, and μ → θ μ → Ker ( θ μ ) equals μ .

Similarly for filters. Thus, we have a bijective correspondence.

Let ˜ S be a fuzzy implicatively closed subset of H and θ = Ψ ˜ S . There exists an order-preserving bijection between: 1.

The set of fuzzy prime ideals of H / θ , and

Let ˜ S be fuzzy implicatively closed, θ = Ψ ˜ S .

Define map F: From fuzzy prime ideals of H / θ to fuzzy prime ideals of H disjoint from ˜ S .

For a fuzzy prime ideal μ ′ of H / θ , define μ ( a ) = μ ′ ( [ a ] θ ) . Then μ is a fuzzy prime ideal of H. For disjointness:

If μ ( a ) ∧ ˜ S ( a ) > 0 , then μ ′ ( [ a ] θ ) ∧ ˜ S ( a ) > 0 . However ˜ S ( a ) ≤ θ ( a , 1 ) (by definition of Ψ ˜ S ), and μ ′ being prime in the quotient implies a contradiction.

Define inverse map G: from fuzzy prime ideals of H disjoint from ˜ S to fuzzy prime ideals of H / θ . For a fuzzy prime ideal μ of H with μ ( a ) ∧ ˜ S ( a ) = 0 for all a , define μ ′ ( [ a ] θ ) = μ ( a ) . This is well defined. If θ ( a , b ) = 1 , then ˜ S ( s ) → θ s ( a , b ) = 1 for all s . This implies μ ( a ) = μ ( b ) by the disjointness condition. Also, μ ′ is prime in H / θ . F and G are order-preserving:

If μ 1 ≤ μ 2 , then clearly F ( μ 1 ) ≤ F ( μ 2 ) . They are inverses: G ( F ( μ ′ ) ) = μ ′ and F ( G ( μ ) ) = μ .

Thus, we have an order-preserving bijection.

Let ˜ S be as in Example 4.2. A fuzzy prime ideal μ is “disjoint” from ˜ S if μ ( x ) ∧ ˜ S ( x ) = 0 for all x .

(First Isomorphism Theorem). Let θ be a fuzzy congruence, μ = Ker ( θ ) , ν = Coker ( θ ) . Then, H / θ ≅ H / μ ≅ H / ν , where H / μ and H / ν are quotients by the fuzzy ideal and filter congruences, respectively.

Let θ be a fuzzy congruence, μ = Ker ( θ ) , ν = Coker ( θ ) . Define ϕ : H / θ → H / μ by ϕ ( [ a ] θ ) = [ a ] μ . This is well defined: If θ ( a , b ) = 1 , then θ ( a , 0 ) = θ ( b , 0 ) by transitivity and symmetry, so μ ( a ) = μ ( b ) . It preserves operations by congruence properties. It is injective: If [ a ] μ = [ b ] μ , then μ ( a ↔ b ) = 1 , so θ ( a , b ) = 1 since θ ( a , b ) ≥ μ ( a ↔ b ) . It is surjective by construction. Define ψ : H / θ → H / ν by ψ ( [ a ] θ ) = [ a ] ν . Similar arguments using Coker ( θ ) show that it is well defined, preserves operations, and is bijective. For any operation ∗ ∈ { ∧ , ∨ , → } : ϕ ( [ a ] θ ∗ [ b ] θ ) = ϕ ( [ a ∗ b ] θ ) = [ a ∗ b ] μ = [ a ] μ ∗ [ b ] μ = ϕ ( [ a ] θ ) ∗ ϕ ( [ b ] θ ) .Thus ϕ and ψ are Heyting algebra isomorphisms, and H / θ ≅ H / μ ≅ H / ν .

With θ from Example 3.2, we can construct H / θ , H / Ker ( θ ) , and H / Coker ( θ ) and show they are isomorphic as fuzzy Heyting algebras.

The set FCong ( H ) of fuzzy congruences on H forms a complete lattice isomorphic to the lattice of fuzzy ideals of H, and the lattice of fuzzy filters of H.

For H = { 0 , a , b , 1 } , the lattice of fuzzy congruences can be partially ordered by pointwise order as θ 1 ≤ θ 2 iff θ 1 ( x , y ) ≤ θ 2 ( x , y ) for all x , y .

(Fuzzy Heyting Topological System). A fuzzy Heyting topological system is a triple ( X , H , ⊢ ) where X is a set, H a Heyting algebra, and ⊢ : X × H → [ 0 , 1 ] a fuzzy satisfaction relation satisfying: ⊢ ( x , a → b ) = in f y ∈ X ( ⊢ ( y , a ) ⇒ ⊢ ( y , b ) ) .

Every fuzzy congruence θ on a Heyting algebra H induces a fuzzy topological system where X = H / θ and ⊢ ( [ x ] , a ) = θ ( x , a ) .