Proposition 1

There exists on E one and only one magma structure for which the mappings $f_\alpha : E_\alpha \to E$ are homomorphisms. If the magmas $E_\alpha$ are associative (resp. commutative), so is E. If the magmas $E_\alpha$ and the homomorphism $f_{\beta\alpha}$ are unital, so are the magma E and the homomorphisms $f_\alpha$.

The magmas $E_\alpha$ will be written multiplicatively.

Let $x, y$ be in E. There exist $\alpha$ in I and $x_\alpha, y_\alpha$ in $E_\alpha$ such that $x = f_\alpha(x_\alpha)$ and $y = f_\alpha(y_\alpha)$. If there exists a magma structure on E for which $f_\alpha$ is a homomorphism, then $x.y = f_\alpha(x_\alpha y_\alpha)$, whence the uniqueness of this magma structure.

To prove the existence, we must prove that for $\alpha, \beta$ in I, $x_\alpha, y_\alpha$ in $E_\alpha$ and $x'_\beta, y'_\beta$ in $E_\beta$, the relations

(5)

imply $f_\alpha(x_\alpha y_\alpha) = f_\beta(x'_\beta y'_\beta)$. For $\gamma \geq \alpha$ and $\gamma \geq \beta$, we set $x_\gamma = f_{\gamma\alpha}(x_\alpha)$, $y_\gamma = f_{\gamma \alpha}(y_\alpha), x'_\gamma = f_{\gamma \beta}(x'_\beta), y'_\gamma = f_{\gamma \beta}(y'_\beta)$. By the definition of direct limit, there exists $\gamma$ in $I$ with $\gamma \geq \alpha, \gamma \geq \beta, x_\gamma = x'_\gamma, y_\gamma = y'_\gamma$. Then Suppose the magmas $E_\alpha$ are associative. Let $x, y, z$ be in $E$. There exist $\alpha \in I$ and elements $x_\alpha, y_\alpha, z_\alpha$ in $E_\alpha$ such that Then $xy = f_\alpha(x_\alpha y_\alpha)$, whence $(xy)z = f_\alpha((x_\alpha y_\alpha)z_\alpha)$; similarly whence $(xy)z = x(yz)$ for $(x_\alpha y_\alpha)z_\alpha = x_\alpha(y_\alpha z_\alpha)$. The case of commutative magmas is treated analogously.

Suppose finally that each magma $E_\alpha$ has an identity element $e_\alpha$ and that $f_{\beta \alpha}(e_\alpha) = e_\beta$ for $\alpha \leq \beta$. For $\alpha, \beta$ in $I$, there exists $\gamma$ in $I$ with $\gamma \geq \alpha$ and $\gamma \geq \beta$, whence

and there thus exists an element $e$ in $E$ such that $f_\alpha(e_\alpha) = e$ for all $\alpha \in I$. Let $x$ be in $E$; choose $\alpha \in I$ and $x_\alpha \in E_\alpha$ such that $x = f_\alpha(x_\alpha)$. Then and similarly $x \cdot e = x$, hence $e$ is the identity element of $E$.

The magma $E$ is called the direct limit of the magmas $E_\alpha$.