A Heyting algebra is an algebra (H,∨,∧,→,0,1) such that:

• 1. (H,∨,∧,0,1) is a bounded distributive lattice.

• 2. for all a,b∈H,a→a=1 .

• 3. b≤a→b.

• 4. a∧(a→b)=a∧b.

• 5. a→(b∧c)=(a→b)∧(a→c).

• 6. (a∨b)→c=(a→c)∧(b→c).

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≤yy,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:

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:

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:

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/ψS and the set of all prime ideals of H disjoint from S, where ψS 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:

Let H={0,a,b,1} as before. Define θ:H×H→[0,1] by:

(x,y)={1ifx=y0.7if{x,y}={0,a}or{0,b}0.5if{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 ˜S be a fuzzy implicatively closed subset of H. Define a fuzzy relation Ψ˜S on H by: Ψ˜S(a,b)=infs∈H(˜S(s)→θs(a,b)) , where θs(a,b)=min(˜S(a→s),˜S(b→s)) and

p→q={1ifp≤qq,otherwise

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)=infs∈H(˜S(s)→θs(a,a)), where θs(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 θs(a,b)=θs(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)→θs(a,b)≥Ψ˜S(a,b)and˜S(s)→θs(b,c)≥Ψ˜S(b,c)⇒˜S(s)→(θs(a,b)∧θs(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:

θs(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))≤θs(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 θμ on H by:

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

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

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

θμ(a,c)∧θμ(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))=θμ(a∧b,c∧d).

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

For the kernel property:

Ker(θμ)(a)=θμ(a,0)=μ(a↔0)=μ(a∧(0→a))=μ(a).

Since a↔0=a in Heyting algebras. Thus θμ is a fuzzy congruence and Ker(θμ)=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, θμ(a,b)=μ(a↔b). Compute a↔b:

a→b=b

b→a=a

a↔b=a∧b=0

Soθμ(a,b)=μ(0)=1 .

Let μ be a fuzzy ideal. The fuzzy congruence generated by μ is:

θμ∗(a,b)=sup{μ(c):c∈H,a↔b≥c}.

With μ as above and a,b with a↔b=0 , then:

θμ∗(a,b)=sup{μ(c):0≥c}=μ(0)=1

θμ∗ 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: θν(a,b)=ν((a→b)∧(b→a)). Then θν is a fuzzy congruence and Coker(θν)=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,so0.6∧0.6=0.6≤0.6 holds.

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

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

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

Since ν(1)=1 and ν(a)≤ν(1).

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

With ν above: θν(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:

(Sketch) Given a fuzzy nucleus j , define θj(a,b)=min(j(a)(b),j(b)(a)) . Conversely, given θ, define jθ(a)(x)=θ(a,a∧x). The verification follows patterns established in.²¹ Building on Hjek’s work on fuzzy logic⁸ and recent developments in fuzzy intuitionistic logic²²:

(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:

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:

With θ from Example 3.2 and a,b:

Ker(θ)(a↔b)=Ker(θ)(0)=1.0,butθ(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)={1ifx=y0.8if(x,y)=(0,a)or(a,0)0.6otherwise

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:

θ→Ker(θ),θ→Coker(θ).

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:

Ifμ(a∧b)=1,thenθ(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 θμ(a,b)=μ(a↔b). We need to show it’s prime:

If θμ(a∧b,0)=1 , then μ(a∧b)=1 . By primeness, μ(a)=1 or μ(b)=1 , so θμ(a,0)=1 or θμ(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 θν as in Theorem 4.19 and verify primeness. The maps are inverses: θ→Ker(θ)→θKer(θ) equals θ, and μ→θμ→Ker(θμ) 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:

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)→θs(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 μ1≤μ2 , then clearly F(μ1)≤F(μ2). 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 θ1≤θ2 iff θ1(x,y)≤θ2(x,y) for all x,y.

Consider a medical diagnosis system where symptoms form a Heyting algebra. Fuzzy congruences can model similarity between symptom patterns, with θ(”fever&cough”,”cough&fatigue”)=0.7 representing 70% similarity between these symptom combinations. Building on Vickers’ work on topological systems¹⁷ and recent fuzzy extensions¹⁵:

(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)=infy∈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) .