Fuzzy Congruences on Heyting Algebras: Characterizations via Fuzzy Ideals and Filters.

Definition 2.1.

A Heyting algebra is an algebra $(H,\vee ,\wedge ,\to ,0,1)$ such that:

Example 2.2.

Let $H=\{0,a,b,1\}$ be a lattice with $0<a<1,0<b<1$ , and $a$ and $b$ incomparable. Define ${}^{“}\to {}^{”}$ as $x\to y=\{\begin{array}{l}1,x\le y\\ y,\mathit{\text{otherwise}}\end{array}$ . Then $(H,\vee ,\wedge ,\to ,0,1)$ is a Heyting algebra. Let H be a nonempty set. A fuzzy subset μ of H is a function μ: H → [0, 1].

Example 2.3.

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

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

Example 2.4.

Let $H$ be as in Example 2.2. Define $\nu :H\to [0,1]$ by $\nu \left(0\right)=0.2,\nu \left(a\right)=0.6,\nu \left(b\right)=0.6,\nu \left(1\right)=1.0$ . Then $\nu$ is fuzzy multiplicatively closed since for any $y\in H$ , $\mathit{min}(\nu \left(x\right),\nu \left(y\right))\le \nu (x\wedge y).$

Definition 2.5.

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

Example 2.6.

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

Definition 2.7.

A fuzzy subset $\nu$ of H is called a fuzzy filter if:

Example 2.8.

Define $\nu :H\to [0,1]$ by:

Then $\nu$ is a fuzzy filter.

Definition 2.9.

A fuzzy ideal $\mu$ is prime if for all $a,b\in H,\mu (a\wedge b)\le \mathit{max}(\mu \left(a\right),\mu \left(b\right)).$

A fuzzy filter ν is prime if for all $a,b\in H,\nu (a\vee b)\le \mathit{max}(\nu \left(a\right),\nu \left(b\right))$ . The foundational work on congruence relations in Heyting algebras by Assaye et al.² established several key results that our fuzzy extensions build upon:

Theorem 2.10.

(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 $H/{\psi }^S$ and the set of all prime ideals of H disjoint from S, where ${\psi }^S$ 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.

Definition 3.1.

Let H be a Heyting algebra. A fuzzy relation $\theta :H\times H\to [0,1]$ is called a fuzzy congruence relation if for all $a,b,c,d\in H:$

Example 3.2.

Let $H=\{0,a,b,1\}$ as before. Define $\theta :H\times H\to [0,1]$ by:

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

Definition 3.3.

(Strong Fuzzy Congruence). A fuzzy relation $\theta :H\times H\to [0,1]$ is a strong fuzzy congruence if it satisfies all conditions of Definition 3.1 plus the additional condition:

Definition 4.1.

Let $˜S$ be a fuzzy implicatively closed subset of H. Define a fuzzy relation ${\Psi }^{˜S}$ on H by: ${\Psi }^{˜S}(a,b)=\mathit{\text{in}}{f}_{s\in H}(˜S\left(s\right)\to {\theta }_s(a,b))$ , where ${\theta }_s(a,b)=\mathit{min}(˜S(a\to s),˜S(b\to s))$ and

Example 4.2.

Let $H=\{0,a,b,1\}$ with $˜S$ defined by:

Then $˜S$ is fuzzy implicatively closed. One can compute ${\Psi }^{˜S}(a,b)$ for some pairs.

Theorem 4.3.

If $˜S$ is a fuzzy implicatively closed subset of $H$ , then ${\Psi }^{˜S}$ is a fuzzy congruence relation on H.

Proof.

Let ˜S be a fuzzy implicatively closed subset of H. We verify each condition: For fuzzy reflexivity: For any $a\in H,{\Psi }^{˜S}(a,a)=\mathit{\text{in}}{f}_{s\in H}(˜S\left(s\right)\to {\theta }_s(a,a)),$ where ${\theta }_s(a,a)=\mathit{min}(˜S(a\to s),˜S(a\to s))=˜S(a\to s).$ Since $˜S\left(s\right)\to ˜S(a\to s)=1$ for all $s$ (by definition of $\to$ in [0,1]), the infimum is reflexive. Fuzzy symmetry is immediate since ${\theta }_s(a,b)={\theta }_s(b,a).$

For fuzzy transitivity, we need to show:

For any $s\in H:$ and Thus:

Since $˜S$ is implicatively closed, we have: $˜S(a\to s)\wedge ˜S(b\to s)\le ˜S((a\to s)\wedge (b\to s))\le ˜S(a\to c)$ when $b\to s\le a\to c.$ This establishes the transitivity condition.

For preservation of ∧, consider for any s:

Using Heyting algebra identity $(a\wedge c)\to s=a\to (c\to s)$ and the fact that ˜S preserves implications, we get:

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

Thus, ${\Psi }^{˜S}$ is a fuzzy congruence relation on H.

Definition 4.4.

Let θ be a fuzzy congruence on H. The fuzzy kernel of θ is: $\mathit{Ker}\left(\theta \right)\left(a\right)=\theta (a,0),\forall a\in H.$ Example 4.5. For the $\theta$ in Example 3.2: $\mathit{Ker}\left(\theta \right)\left(0\right)=\theta (0,0)=1,\mathit{Ker}\left(\theta \right)\left(a\right)=\theta (a,0)=0.7,\mathit{Ker}\left(\theta \right)\left(b\right)=0.7,\mathit{Ker}\left(\theta \right)\left(1\right)=0.3.$

Theorem 4.6.

If θ is a fuzzy congruence on H, then $\mathit{Ker}\left(\theta \right)$ is a fuzzy ideal of H.

Proof.

Let θ be a fuzzy congruence on H. First, $\mathit{Ker}\left(\theta \right)\left(0\right)=\theta (0,0)=1$ by reflexivity.

Second, for $a,b\in H:\mathit{Ker}\left(\theta \right)\left(a\right)\wedge \mathit{Ker}\left(\theta \right)\left(b\right)=\theta (a,0)\wedge \theta (b,0)$ . By the preservation of ∨ (condition 5 in Definition 3.1) $\theta (a,0)\wedge \theta (b,0)\le \theta (a\vee b,0\vee 0)=\theta (a\vee b,0)=\mathit{Ker}\left(\theta \right)(a\vee b)$ . Third, for any $b\in H:\mathit{Ker}\left(\theta \right)\left(a\right)=\theta (a,0)\le \theta (a\wedge b,0\wedge b)=\theta (a\wedge b,0)=\mathit{Ker}\left(\theta \right)(a\wedge b)$ using preservation of ∧ and the fact that $0\wedge b=0.$

Thus $,\mathit{Ker}\left(\theta \right)$ satisfies all conditions of a fuzzy ideal.

Theorem 4.7.

Let μ be a fuzzy ideal of H. Define a fuzzy relation ${\theta }_{\mu }$ on H by:

${\theta }_{\mu }(a,b)=\mu (a\leftrightarrow b)$ , where $a\leftrightarrow b=(a\to b)\wedge (b\to a).$ Then, ${\theta }_{\mu }$ is a fuzzy congruence on H and $\mathit{Ker}\left(\mathit{\theta \mu }\right)=\mu .$

Proof.

Let $\mu$ be a fuzzy ideal. Define ${\theta }_{\mu }(a,b)=\mu (a\leftrightarrow b)$ where $a\leftrightarrow b=(a\to b)\wedge (b\to a).$

First, we show ${\theta }_{\mu }$ is a fuzzy congruence: Reflexivity: ${\theta }_{\mu }(a,a)=\mu (a\leftrightarrow a)=\mu \left(1\right)=1$ since $\mu \left(1\right)=\mu (0\to 0)\ge \mu \left(0\right)=1.$ Symmetry: ${\theta }_{\mu }(a,b)=\mu (a\leftrightarrow b)=\mu (b\leftrightarrow a)={\theta }_{\mu }(b,a).$

Transitivity: We need $\mu (a\leftrightarrow b)\wedge \mu (b\leftrightarrow c)\le \mu (a\leftrightarrow c).$

In any Heyting algebra,

Since μ is a fuzzy ideal:

Preservation of $\wedge :$

In Heyting algebras, $(a\leftrightarrow c)\wedge (b\leftrightarrow d)\le (a\wedge b)\leftrightarrow (c\wedge d),\mathit{so}:$

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

For the kernel property:

Since $a\leftrightarrow 0=a$ in Heyting algebras. Thus ${\theta }_{\mu }$ is a fuzzy congruence and $\mathit{Ker}\left({\theta }_{\mu }\right)=\mu .$

Example 4.8.

Let $H=\{0,a,b,1\}$ with $\mu$ as in Example 2.6:

Then, ${\theta }_{\mu }(a,b)=\mu (a\leftrightarrow b).$ Compute $a\leftrightarrow b:$

$\mathrm{So}{\mathrm{\theta }}_{\mathrm{\mu }}(\mathrm{a},\mathrm{b})=\mathrm{\mu }\left(0\right)=1$ .

Definition 4.9.

Let $\mu$ be a fuzzy ideal. The fuzzy congruence generated by $\mu$ is:

Example 4.10.

With $\mu$ as above and $a,b$ with $a\leftrightarrow b=0$ , then:

Theorem 4.11.

${\theta }_{\mu }^{\ast }$ is the smallest fuzzy congruence on H whose kernel contains μ.

Definition 4.12.

Let $\theta$ be a fuzzy congruence on $H$ . The fuzzy cokernel is:

Example 4.13.

For θ from Example 3.2:

Theorem 4.14.

If θ is a fuzzy congruence, then $\mathit{\text{Coker}}\left(\theta \right)$ is a fuzzy filter of H.

Proof.

Let θ be a fuzzy congruence. First, $\mathit{\text{Coker}}\left(\theta \right)\left(1\right)=\theta (1,1)=1.$

Second, for $b\in H:$

Third, for any $b\in H:$

Thus, $\mathit{\text{Coker}}\left(\theta \right)$ satisfies all conditions of a fuzzy filter.

Theorem 4.15.

(Characterization via Cokernel Filter). Let $\nu$ be a fuzzy filter of $H$ .

Define: ${\theta }_{\nu }(a,b)=\nu ((a\to b)\wedge (b\to a)).$ Then ${\theta }_{\nu }$ is a fuzzy congruence and $\mathit{\text{Coker}}\left({\theta }_{\nu }\right)=\nu .$

Example 4.16.

Let $\nu$ be as in Example 2.8. Then:

Definition 4.17.

A fuzzy filter ν is implicative if: $\nu (a\to b)\wedge \nu \left(a\right)\le \nu \left(b\right).$

Example 4.18.

Define $\nu :H\to [0,1]\mathit{by}:\nu \left(0\right)=0.4,\nu \left(a\right)=0.6,\nu \left(b\right)=0.6,\nu \left(1\right)=1$

Check: $\nu (a\to b)=\nu \left(b\right)=0.6,\nu \left(a\right)=0.6,\nu \left(b\right)=0.6,\mathit{so}0.6\wedge 0.6=0.6\le 0.6$ holds.

Theorem 4.19.

For a fuzzy implicative filter $\nu ,$ the relation: ${\theta }_{\nu }(a,b)=\nu (a\to b)\wedge \nu (b\to a)$ is a fuzzy congruence, and $\nu =\mathit{\text{Coker}}\left({\theta }_{\nu }\right).$

Proof.

Let ν be a fuzzy implicative filter. Define ${\theta }_{\nu }(a,b)=\nu (a\to b)\wedge \nu (b\to a).$

Reflexivity: ${\theta }_{\nu }(a,a)=\nu (a\to a)\wedge \nu (a\to a)=\nu \left(1\right)\wedge \nu \left(1\right)=1.$

Symmetry is immediate from the definition.

Transitivity requires $\nu (a\to b)\wedge \nu (b\to a)\wedge \nu (b\to c)\wedge \nu (c\to b)\le \nu (a\to c)\wedge \nu (c\to a)$ from implicative filter property: $\nu (a\to b)\wedge \nu (b\to c)\le \nu (a\to c).$

Similarly: $\nu (c\to b)\wedge \nu (b\to a)\le \nu (c\to a).$

Thus, the inequality holds. Preservation of operations follows from Heyting algebra identities and filter properties. For the cokernel:

Since $\nu \left(1\right)=1$ and $\nu \left(a\right)\le \nu \left(1\right).$

Thus ${\theta }_{\nu }$ is a fuzzy congruence and $\nu =\mathit{\text{Coker}}\left({\theta }_{\nu }\right).$

Example 4.20.

With $\nu$ above: ${\theta }_{\nu }(a,b)=\nu (a\to b)\wedge \nu (b\to a)=\nu \left(b\right)\wedge \nu \left(a\right)=0.6\wedge 0.6=0.6.$ Recent work by Zhao et al.²¹ on fuzzy nuclei in residuated lattices inspires the following connection:

Theorem 4.21.

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

Proof.

(Sketch) Given a fuzzy nucleus $j$ , define ${\theta }_j(a,b)=\mathit{min}(j\left(a\right)\left(b\right),j\left(b\right)\left(a\right))$ . Conversely, given $\theta ,$ define $j_{\theta }\left(a\right)\left(x\right)=\theta (a,a\wedge x).$ The verification follows patterns established in.²¹ Building on Hjek’s work on fuzzy logic⁸ and recent developments in fuzzy intuitionistic logic²²:

Definition 4.22.

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

$\theta (a,b)\le \theta (\neg \neg a,\neg \neg b)$ where $\neg x=x\to 0$ is the intuitionistic negation

Theorem 4.23.

Every fuzzy congruence θ on a Heyting algebra H induces a fuzzy congruence ${\theta }^{\ast \ast }$ on the Boolean algebra $H^{\ast \ast }=\{a^{\ast \ast }:a\in H\}$ via $:$ ${\theta }^{\ast \ast }(a^{\ast \ast },b^{\ast \ast })=\mathit{sup}\{\theta (x,y):x^{\ast \ast }=a^{\ast \ast },y^{\ast \ast }=b^{\ast \ast }\}$ .

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 $H/\theta$ as the set of fuzzy equivalence classes ${\left[a\right]}_{\theta }$ , where the membership degree of $x$ in ${\left[a\right]}_{\theta }$ is $\theta (a,x).$

Example 4.24.

For θ from Example 3.2, the equivalence classes are:

Theorem 4.25.

The fuzzy quotient set $H/\theta$ can be equipped with operations $\wedge ,\vee ,\to$ such that:

Then $H/\theta$ forms a Heyting algebra in the fuzzy sense, called the fuzzy quotient Heyting algebra.

Theorem 4.26.

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

Example 4.27.

With $\theta$ from Example 3.2 and $a,b:$

This shows the equality may require specific definitions of $\leftrightarrow$ in the fuzzy context.

Definition 4.28.

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

Example 4.29.

Define θ by:

For $a\wedge b=0,\theta (a\wedge b,0)=\theta (0,0)=1$ . We need $\theta (a,0)=1$ or $\theta (b,0)=1,$ but both are $0.8$ , so not prime.

Theorem 4.30.

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

Proof.

We establish the bijection between prime fuzzy congruence and fuzzy prime ideals. Given a prime fuzzy congruence $\theta ,\mathit{Ker}\left(\theta \right)$ is a fuzzy ideal by Theorem 4.6. For the prime condition:

By primness of $\theta$ , either $\theta (a,0)=1$ or $\theta (b,0)=1$ , so $\mu \left(a\right)=1$ or $\mu \left(b\right)=1$ . Conversely, given a fuzzy prime ideal $\mu$ , define ${\theta }_{\mu }(a,b)=\mu (a\leftrightarrow b).$ We need to show it’s prime:

If ${\theta }_{\mu }(a\wedge b,0)=1$ , then $\mu (a\wedge b)=1$ . By primeness, $\mu \left(a\right)=1$ or $\mu \left(b\right)=1$ , so ${\theta }_{\mu }(a,0)=1$ or ${\theta }_{\mu }(b,0)=1$ . For the correspondence with fuzzy prime filters:

Given $\theta ,\mathit{\text{Coker}}\left(\theta \right)$ is a fuzzy filter by Theorem 4.14. For primeness:

If $\nu (a\vee b)=1$ , then $\theta (a\vee b,1)=1.$ By congruence properties and primeness, either $\theta (a,1)=1$ or $\theta (b,1)=1$ . Conversely, given a fuzzy prime filter $\nu$ , define ${\theta }_{\nu }$ as in Theorem 4.19 and verify primeness. The maps are inverses: $\theta \to \mathit{Ker}\left(\theta \right)\to {\theta }_{\mathit{Ker}\left(\theta \right)}$ equals θ, and $\mu \to {\theta }_{\mu }\to \mathit{Ker}\left({\theta }_{\mu }\right)$ equals $\mu .$

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

Theorem 4.31.

Let $˜S$ be a fuzzy implicatively closed subset of $H$ and $\theta ={\Psi }^{˜S}$ . There exists an order-preserving bijection between:

Proof.

Let $˜S$ be fuzzy implicatively closed, $\theta ={\Psi }^{˜S}$ .

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

For a fuzzy prime ideal ${\mu }^{\prime }$ of $H/\theta$ , define $\mu \left(a\right)={\mu }^{\prime }\left({\left[a\right]}_{\theta }\right).$ Then μ is a fuzzy prime ideal of H. For disjointness:

If $\mu \left(a\right)\wedge ˜S\left(a\right)>0$ , then ${\mu }^{\prime }\left({\left[a\right]}_{\theta }\right)\wedge ˜S\left(a\right)>0$ . However $˜S\left(a\right)\le \theta (a,1)$ (by definition of ${\Psi }^{˜S}$ ), and ${\mu }^{\prime }$ 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 $H/\theta$ . For a fuzzy prime ideal $\mu$ of H with $\mu \left(a\right)\wedge ˜S\left(a\right)=0$ for all $a$ , define ${\mu }^{\prime }\left({\left[a\right]}_{\theta }\right)=\mu \left(a\right)$ . This is well defined. If $\theta (a,b)=1$ , then $˜S\left(s\right)\to {\theta }_s(a,b)=1$ for all $s$ . This implies $\mu \left(a\right)=\mu \left(b\right)$ by the disjointness condition. Also, ${\mu }^{\prime }$ is prime in $H/\theta$ . $F$ and $G$ are order-preserving:

If ${\mu }_1\le {\mu }_2$ , then clearly $F\left({\mu }_1\right)\le F\left({\mu }_2\right).$ They are inverses: $G\left(F\left({\mu }^{\prime }\right)\right)={\mu }^{\prime }$ and $F\left(G\left(\mu \right)\right)=\mu$ .

Thus, we have an order-preserving bijection.

Example 4.32.

Let $˜S$ be as in Example 4.2. A fuzzy prime ideal $\mu$ is “disjoint” from $˜S$ if $\mu \left(x\right)\wedge ˜S\left(x\right)=0$ for all $x$ .

Theorem 4.33.

(First Isomorphism Theorem). Let θ be a fuzzy congruence, $\mu =\mathit{Ker}\left(\theta \right)$ , $\nu =\mathit{\text{Coker}}\left(\theta \right)$ . Then, $H/\theta \cong H/\mu \cong H/\nu ,$ where $H/\mu$ and $H/\nu$ are quotients by the fuzzy ideal and filter congruences, respectively.

Proof.

Let $\theta$ be a fuzzy congruence, $\mu =\mathit{Ker}\left(\theta \right),\nu =\mathit{\text{Coker}}\left(\theta \right).$ Define $\varphi :H/\theta \to H/\mu$ by $\varphi \left({\left[a\right]}_{\theta }\right)={\left[a\right]}_{\mu }$ . This is well defined: If $\theta (a,b)=1$ , then $\theta (a,0)=\theta (b,0)$ by transitivity and symmetry, so $\mu \left(a\right)=\mu \left(b\right)$ . It preserves operations by congruence properties. It is injective: If ${\left[a\right]}_{\mu }={\left[b\right]}_{\mu }$ , then $\mu (a\leftrightarrow b)=1$ , so $\theta (a,b)=1$ since $\theta (a,b)\ge \mu (a\leftrightarrow b)$ . It is surjective by construction. Define $\psi :H/\theta \to H/\nu$ by $\psi \left({\left[a\right]}_{\theta }\right)={\left[a\right]}_{\nu }.$ Similar arguments using $\mathit{\text{Coker}}\left(\theta \right)$ show that it is well defined, preserves operations, and is bijective. For any operation $\ast \in \{\wedge ,\vee ,\to \}:\varphi ({\left[a\right]}_{\theta }\ast {\left[b\right]}_{\theta })=\varphi \left({[a\ast b]}_{\theta }\right)={[a\ast b]}_{\mu }={\left[a\right]}_{\mu }\ast {\left[b\right]}_{\mu }=\varphi \left({\left[a\right]}_{\theta }\right)\ast \varphi \left({\left[b\right]}_{\theta }\right)$ .Thus $\varphi$ and $\psi$ are Heyting algebra isomorphisms, and $H/\theta \cong H/\mu \cong H/\nu .$

Example 4.34.

With θ from Example 3.2, we can construct $H/\theta ,H/\mathit{Ker}\left(\theta \right)$ , and $H/\mathit{\text{Coker}}\left(\theta \right)$ and show they are isomorphic as fuzzy Heyting algebras.

Theorem 4.35.

The set $\mathit{\text{FCong}}\left(H\right)$ 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.

Example 4.36.

For $H=\{0,a,b,1\}$ , the lattice of fuzzy congruences can be partially ordered by pointwise order as ${\theta }_1\le {\theta }_2$ iff ${\theta }_1(x,y)\le {\theta }_2(x,y)$ for all $x,y.$

Example 5.1.

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

Definition 5.2.

(Fuzzy Heyting Topological System). A fuzzy Heyting topological system is a triple $(X,H,⊢)$ where X is a set, H a Heyting algebra, and $⊢:X\times H\to [0,1]$ a fuzzy satisfaction relation satisfying: $⊢(x,a\to b)=\mathit{\text{in}}{f}_{y\in X}(⊢(y,a)\Rightarrow ⊢(y,b))$ .

Theorem 5.3.

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