Second, We Consider Σ-Algebras Freely Generated Relative to a Subcategory Alg(Σ, *CE*)

Let *CE* be a set of conditional Σ-equations. A Σ-algebra F = (*F*, Ω<sup>F</sup> ) is freely generated in Alg(Σ, *CE*) by an Eq-sketch G = (*X*, *R*) if, and only if, F |= *CE*, and there is a valid interpretation (*η*G, F) of G in F that is universal relative to Alg(Σ, *CE*). That is, for all Σ-algebras A = (*A*, ΩA) in Alg(Σ, *CE*) and all valid interpretations (*ι*, A) of G in A there exists a unique morphism *ι* ◦ : F→A such that *η*G; *ι* ◦ = *ι* in Base*EQ*, i.e., such that *ι* ◦ establishes an interpretation morphism *ι* ◦ : (*η*G, F) → (*ι*, A) in Int(G) Int(*X*).

$$\mathsf{Int}(\mathbb{G}) \downarrow \mathsf{Alg}(\Sigma, \mathsf{CE}) \qquad \qquad \underbrace{\mathrm{X} \xrightarrow{(\eta \Box, \mathcal{F}) \vdash \times \mathcal{R}} \mathrm{F}}\_{(\iota, \mathcal{A}) \vdash\_{\mathcal{X}} \mathcal{R}} \mathrm{F} \qquad \qquad \begin{array}{c} \mathcal{F} = (\mathsf{F}, \Omega^{\mathsf{F}}) \qquad \qquad \mathsf{Alg}(\Sigma, \mathsf{CE}) \vdash\_{\mathcal{X}} \mathcal{R} \\\\ \downarrow \\ \mathcal{A} = (\mathsf{A}, \Omega^{\mathsf{A}}) \end{array}$$

In this case, the universal property of (*η*G, F) entails that (*η*G, F) is initial in the subcategory Int(G) <sup>↓</sup> Alg(Σ, *CE*) = <sup>Π</sup>−<sup>1</sup> *<sup>G</sup>* (Alg(Σ, *CE*)) of Int(G) given by all valid interpretations of G in Σ-algebras in Alg(Σ, *CE*). Moreover, we obtain an isomorphism between Int(G) ↓ Alg(Σ, *CE*) and the co-slice category F/Alg(Σ, *CE*).

For arbitrary sets *CE* of conditional Σ-equations and arbitrary Eq-sketches G = (*X*, *R*), a Σ-algebra F, freely generated by G in Alg(Σ, *CE*), exists and is uniquely determined "up

to isomorphism". In the introductory Section 1.1.1, we used the notation F(Σ, *CE*, *X*, *R*) to denote those freely generated Σ-algebras. F(Σ, *CE*, *X*, *R*) can be constructed as a quotient of TΣ(*X*).

In case of groups, *CE* is a set of conditional Σ-equations with an empty premise, representing the group axioms, and F(Σ, *CE*, *X*, *R*) is called the group freely generated by the set of generators *X* and the set *R* of defining relations.
