Example

(1) Take $H = \{1, 2, 3\}$; the algebras which admit

as universal relator are the associative algebras. The algebras which admit $X_1 X_2 - X_2 X_1$ as universal relator are the commutative algebras. The algebras which admit the universal relators $X_1 X_1$ and

$$(X_1 X_2) X_3 + (X_2 X_3) X_1 + (X_3 X_1) X_2$$

as universal relator are the Lie algebras.

Let I be a set; let a family $(S_k)_{k \in K}$ of elements of Lib_A(H) be given and consider the set T of elements of Lib_A(I) of the form $S_k((U_h))_{h \in H}$, where k runs through K and, for each k, $(U_h)_{h \in H}$ runs through the set of families of elements of Lib_A(I) with H as indexing set; consider a family $(R_j)_{j \in J}$ with T as set of elements. Then let F be the universal algebra defined by the generating system I related by the family $(R_j)_{j \in J}$ and let $u : \mathrm{Lib}_A(I) \to F$ be the canonical homomorphism, so that Ker(u) is generated by the elements $S_k((U_h)_{h \in H})$ for all $k \in K$ and all families $(U_h)_{h \in H}$ of elements of Lib_A(I); clearly each of the $S_k$ ($k \in K$) is a universal relator for F. Now let F' be an algebra admitting a generating system $(x_i)_{i \in I}$, for which each of the $S_k$ is a universal relator, and let $u' : \mathrm{Lib}_A(I) \to F'$ be the homomorphism such that $u'(X_i) = x_i$ for all $i \in I$; clearly Ker(u) $\subset$ Ker(u') and hence $u'$ can be written uniquely in the form $u' = h \circ u$, where $h : F \to F'$ is a homomorphism such that $h(\overline{X}_i) = x_i$ for all $i \in I$. For this reason F is called the universal algebra defined by the generating system I, corresponding to the family of universal relators $(S_k)_{k \in K}$. By an abuse of language, F is sometimes called the universal algebra generated by I and subject to the identities $S_k((u_h)) = 0$ for every family $(u_h)_{h \in H}$ of elements of F.

Example 2. Let L' be the universal algebra generated by I and subject to the identities $(uv)w - u(vw) = 0$ for every family of three elements of L' and let L'' be the algebra obtained by adjoining a unit element to L'; then there exists a unique unital isomorphism g of L'' onto Libas_A(I) such that $g(\overline{X}_i) = X_i$ for all $i \in I$. For clearly L'' is associative and the existence of the homomorphism g follows from the definition of L' and the remarks preceding it; but then clearly L'' satisfies the universal property (no. 7, Proposition 7) which characterizes Libas_A(I), whence the conclusion.

Considerations analogous to the above can be applied to associative algebras (resp. commutative associative algebras), taking account of the following remarks. When the context gives sufficient indication that the algebras considered are unital algebras, by an abuse of language, a family of elements $(x_i)_{i \in I}$ of an algebra $F$, such that the subalgebra generated by the $x_i$ ($i \in I$) and the unit element is equal to $F$, is often called a generating system of $F$. Then let $F$ be a unital associative algebra over $A$ and let $(x_i)_{i \in I}$ be a family of elements of $F$; by virtue of no. 7, Proposition 7, there exists a unique unital homomorphism $f : \mathrm{Libas}_A(I) \to F$ such that $f(X_i) = x_i$ for all $i \in I$; if $U \in \mathrm{Libas}_A(I)$, $f(U)$ is also called the element of $F$ derived from $U$ by substituting the elements $x_i$ for the indeterminates $X_i$ and it is also denoted by $U((x_i)_{i \in I})$. Then the notions of relator, presentation and universal relator go over immediately to associative algebras; it suffices simply to replace $\mathrm{Lib}_A(I)$ throughout by $\mathrm{Libas}_A(I)$. The universal unital associative algebra defined by the generating system I related by the family of relators $(R_j)_{j \in J}$ is the quotient algebra of $\mathrm{Libas}_A(I)$ by the two-sided ideal generated by the family $(R_j)$. The universal unital associative algebra generated by the generating system $I$, corresponding to a family of universal relators is defined similarly. We leave to the reader the task of stating the analogous definitions relative to commutative associative algebras with $\mathrm{Libasc}_A(I)$ substituted for $\mathrm{Libas}_A(I)$.