*4.3. Satisfaction Relation and Satisfaction Condition*

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*A*

**Definition 15** (Satisfaction relation for equations)**.** *For any context K* ∈ Cxt*EQ, any* Σ*-equation* (*K*, *t*<sup>1</sup> = *t*2) *in K and any interpretation* (*ι*, A) *of context K in a* Σ*-algebra* A = (*A*, ΩA)*, we define:*

$$\begin{aligned} \left(\iota,\mathcal{A}\right) \vdash\_{\mathcal{K}} \left(\mathcal{K},t\_{1}=t\_{2}\right) \quad \text{iff} \quad \iota^{\diamond}(t\_{1}) = \iota^{\diamond}(t\_{2}) \quad \text{(i.e. } \ t\_{1}^{\mathcal{A}}(\iota) = t\_{2}^{\mathcal{A}}(\iota)\text{)}\\ \longleftrightarrow\_{\iota^{\square}} \sum\_{\iota^{\square}}^{\subseteq} T\_{\mathcal{E}}(\mathcal{K}) \end{aligned} \tag{9}$$

The satisfaction condition is ensured by the well-behaved interplay of translations of terms along context morphisms and evaluations of terms.

**Proposition 1** (Satisfaction condition for equations)**.** *For any morphism ϕ*: *K* → *G in* Cxt*EQ, any* Σ*-equation* (*K*, *t*<sup>1</sup> = *t*2) *in K and any interpretation* (, A) *of context G in a* Σ*-algebra* A = (*A*, ΩA)*, we have:*

**Proof.** Due to the definition of the functors Int : Cxt*op EQ* → Cat, Eq : Cxt*EQ* → Set and the fact that (*ϕ*; )◦ = *ϕ*∗; ◦, we obtain the commutative diagram, above on the right, thus the satisfaction condition follows immediately from Definition 15.

Summarizing all definitions and results, we obtain the main result in this section:

**Proposition 2** (Institution of Equations)**.** *Any choice of a finite set S and a signature* Σ = (Ω, *in*, *out*) *establishes a corresponding Institution of Equations* IE = (Cxt*EQ*, Eq, Int, |=)*.*
