Mathematics, Woldia University, Woldia, Amhara, 400, Ethiopia

DEREBEW NIGUSSIE DERSO
Roles: Conceptualization, Formal Analysis, Methodology, Software, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

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 4.16.

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 3.5. 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

S0,θμ(a,b)=μ(0)=1 . Hence, θμ is a fuzzy congruence on H and Ker(θμ)=μ.

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.

Hence, the fuzzy cokernel is verified by this example.

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.

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

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 as defined in Example 4.4 (and satisfying the implicative filter property from Theorem 4.5) above, we compute 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 3.6. 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 4.16. 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 4.5 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) .

No datasets were generated or analyzed during this study. All results are derived analytically, and all supporting information is fully contained within the manuscript.

The authors acknowledge the referees for their excellent academic support.

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PUBLISHED 23 Jun 2026

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MUKHTAR AHMAD, Institut Teknologi Bandung, Bandung, Indonesia 

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Raghad I. Sabri, University of Technology, Baghdad, Iraq 

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