Definition 2

A direct system of magmas relative to the indexing set I is a direct system of sets $(E_\alpha, f_{\beta\alpha})$ relative to I, each $E_\alpha$ having a magma structure and each $f_{\beta\alpha}$ being a magma homomorphism.

Let $(E_\alpha, f_{\beta\alpha})$ be a direct system of magmas. E will denote the direct limit set $\lim_{\longrightarrow} E_\alpha$ and $f_\alpha$ the canonical mapping of $E_\alpha$ into E. Recall that

(2)

(3)

By (2), also

(4)

If $x_\alpha, y_\alpha \in E_\alpha$ are such that $f_\alpha(x_\alpha) = f_\alpha(y_\alpha)$, there exists a $\beta \geq \alpha$ such that $f_{\beta\alpha}(x_\alpha) = f_{\beta\alpha}(y_\alpha)$.