**4. Institutions of Equations**

With this section about Institutions of Equational Statements, or short Institutions of Equations, we start to close the circle to the ideas and motivations discussed in the introductory Section 1.1.1 Universal Algebra and Algebraic Specifications. In these areas, substitutions play a central role and, analyzing the situation in these areas, we may obtain also some hints and guidelines for the future development of a more abstract and general account of substitutions in Logics of Statements.

Equations are the main conceptual tool in Universal Algebra. To define equational statements, we could again apply the encapsulation trick we have used in the last subsections to define statements for footprints with feature symbols only. That is, we could introduce atomic equations *X t*<sup>1</sup> = *t*2, define atomic equational statements (*X*, *t*<sup>1</sup> = *t*2, *γ*) in contexts *K* with *γ* : *X* → *K* and translate atomic equational statements along context morphisms by simple post-composition.

This idea works fine as long as we are only interested in formalisms to describe and specify algebraic structures. The encapsulation approach seems to be not appropriate, however, to describe and work with instances of equations w.r.t. substitutions of variables by terms. The construction of those instances is a crucial tool in any deduction calculus in Universal Algebra; thus, we decided to work instead of the encapsulation-based two-step approach with a one-step approach defining directly equations *K t*<sup>1</sup> = *t*<sup>2</sup> in contexts *K*.

This means that we adapt for Institutions of Equations the construction scheme in Figure 1 in the following way: We have Str(Ξ) = Sem(Ξ). Step (6) is dropped and we construct directly Stm(*K*) as a set of equations in context *K*. Correspondingly, the satisfaction relations |=*<sup>K</sup>* are defined by means of the evaluation of terms in algebras.

As a complement to the FOL-example, we consider many-sorted total algebras and conditional equations. In this section, we define corresponding Institutions of Equations while conditional equations are formalized and discussed in Section 5.3.

In accordance with the FOL-example, we fix a finite set *S* ∈ Set*Obj* of sort symbols and choose as Base*EQ* the interpretation category Set*<sup>S</sup>* = [*<sup>S</sup>* <sup>→</sup> Set]. Var*EQ* is the full subcategory of Base*EQ* given by all finite and disjoint *S*-sets *X* = (*Xs* | *s* ∈ *S*) with *Xs* a subset of the set {*x*, *x*1, *x*2,..., *y*, *y*1, *y*2,...} for all *s* ∈ *S*.
