*5.3. Sketches of Equations*

For an Institutions of Equations IE = (Cxt*EQ*, Eq, Int, |=) an Eq-sketch E = (*X*, *E*) is given by a context *X* in Cxt*EQ*, i.e., an *S*-set *X*, and a set *E* of Σ-equations in *X*. A valid interpretation of E = (*X*, *E*) is an interpretation (*ι*, A) of context *X* in a Σ-algebra A = (*A*, ΩA) such that (*ι*, A) |=*<sup>X</sup> E*, i.e., (*ι*, A) |=*<sup>X</sup>* (*X*, *t*<sup>1</sup> = *t*2) for all Σ-equations (*X*, *t*<sup>1</sup> = *t*2) in *E* according to (9).

Based on these definitions, we can define IE-morphisms, Eq-sketch arrows and IEimplications, respectively, exactly in the same way as we have done it for Institutions of Statements IS = (Cxt, Stm, Int, |=) in Sections 5.1 and 5.2. Moreover, we have, obviously, for Institutions of Equations also corresponding variants of Definition 22 (Validity of Sketch Implications), Definition 24 (Entailment of Sketch Implications), Lemma 1 (Factorization of Sketch Implications) and Theorem 2 (Semantic Deduction Theorem) available.

Abstract and Universal Algebra have been developed independent of First-Order Logic and conditional Σ-equations are usually not introduced as "universally quantified implications". They are rather described as IE-implications (*Y*, *Prem*) ⇒ (*Y*, *Conc*), in the sense of Definition 21 where *Prem* represents the set of equations in the premise of a conditional Σ-equation and *Conc* the single equation in the conclusion. In particular, the validity of conditional Σ-equations in Σ-algebras A is defined in perfect accordance with Definition 22 (Validity of Sketch Implications) (compare [6,7,41]). Therefore, we will also use the term **conditional** Σ**-equation** for IE-implications (*Y*, *Prem*) ⇒ (*Y*, *Conc*) with *Y*, *Prem* finite and *Conc* a singleton.

Finally, we reached the point where we can give an answer to Question 2 (p. 3): Yes, Theorem 2 is the general Semantic Deduction Theorem, we have been looking for and the equivalence, mentioned in the question, corresponds to the specialization of the general Semantic Deduction Theorem for conditional Σ-equations.

#### 5.3.1. Freely Generated Algebras

The footprints in Institutions of Equations are algebraic signatures Σ = (Ω, *in*, *out*) and we have Str(Σ) = Sem(Σ) := Alg(Σ). Conditional Σ-equations are the traditional means to specify subcategories of Alg(Σ). Given a set *CE* of Conditional Σ-equations, we denote by Alg(Σ, *CE*) the subcategory of Alg(Σ) given by all those Σ-algebras A such that A |= *CE*, i.e., A |= P ⇒ C (as defined in Definition 22) for all conditional Σ-equations P ⇒ C in *CE* (compare Remark 27).

In case P = (*Y*, ∅), we may call P ⇒ C a conditional Σ-equation with an empty premise. Note that there is a simply but crucial conceptual difference between a Σ-equation (*Y*, *t*<sup>1</sup> = *t*2) and the corresponding conditional Σ-equation (*Y*, ∅) ⇒ (*Y*, {(*Y*, *t*<sup>1</sup> = *t*2)}) with an empty premise. (*Y*, *t*<sup>1</sup> = *t*2) is just a simple *statement in context Y* while (*Y*, ∅) ⇒ (*Y*, {(*Y*, *t*<sup>1</sup> = *t*2)}) is a tool to make statements about Σ-algebras. Being not aware of this difference is often a source of confusion!
