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An immediate generalization of the concept of the direct limit of a direct family of sets is the direct limit of a direct family of algebraic systems.
The definition is almost identical to that of a direct family of sets, except that functions $\phi_{ij}$ are now homomorphisms. For completeness, we will spell out the definition in its entirety.
Let $\mathcal{A}=\lbrace A_i\mid i\in I\rbrace$ be a family of algebraic systems of the same type (say, they are all $O$ -algebras), indexed by a non-empty set $I$ . $\mathcal{A}$ is said to be a direct family if
- $I$ is a directed set,
- whenever $i\le j$ in $I$ , there is a homomorphism $\phi_{ij}:A_i\to A_j$ ,
- $\phi_{ii}$ is the identity on $A_i$ ,
- if $i\le j\le k$ , then $\phi_{jk}\circ \phi_{ij}=\phi_{ik}$ .
An example of this is a direct family of sets. A homomorphism between two sets is just a function between the sets.
Let $\mathcal{A}$ be a direct family of algebraic systems $A_i$ , indexed by $I$ ($i\in I$ ). Take the disjoint union of the underlying sets of each algebraic system, and call it $A$ . Next, a binary relation $\sim$ is defined on $A$ as follows:
given that $a\in A_i$ and $b\in A_j$ , $a\sim b$ iff there is $A_k$ such that $\phi_{ik}(a)=\phi_{jk}(b)$ .
It is shown here that $\sim$ is an equivalence relation on $A$ , so we can take the quotient $A/\sim$ , and denote it by $A_{\infty}$ . Elements of $A_{\infty}$ are denoted by $[a]_I$ or $[a]$ when there is no confusion, where $a\in A$ . So $A_{\infty}$ is just the direct limit of $A_i$ considered as
sets.
Next, we want to turn $A_{\infty}$ into an $O$ -algebra. Corresponding to each set of $n$ -ary operations $\omega_i$ defined on $A_i$ for all $i\in I$ , we define an $n$ -ary operation $\omega$ on $A_{\infty}$ as follows:
for $i=1,\ldots,n$ , pick $a_i\in A_{j(i)}$ , $j(i)\in I$ . Let $J:=\lbrace j(i)\mid i=1,\ldots,n\rbrace$ . Since $I$ is directed and $J$ is finite, $J$ has an upper bound $j\in I$ . Let $\alpha_i=\phi_{j(i)j}(a_i)$ . Define $$\omega([a_1],\ldots,[a_n]):=[\omega_j(\alpha_1,\ldots,\alpha_n)].$$
Proposition 1 $\omega$ is a well-defined $n$ -ary operation on $A_{\infty}$ .
Proof. Suppose $[b_1]=[a_1],\ldots, [b_n]=[a_n]$ . Let $\alpha_i$ be defined as above, and let $a:=\omega_j(\alpha_1,\ldots,\alpha_n)\in A_j$ . Similarly, $\beta_i$ are defined: $\beta_i:=\phi_{k(i)k}(b_i)\in A_k$ , where $b_i\in A_{k(i)}$ . Let $b:=\omega_k(\beta_1,\ldots,\beta_n)\in A_k$ . We want to show that $a\sim b$ .
Since $a_i\sim b_i$ , $\alpha_i\sim \beta_i$ . So there is $c_i:= \phi_{j\ell(i)}(\alpha_i)=\phi_{k\ell(i)}(\beta_i)\in A_{\ell(i)}$ . Let $\ell$ be the upper bound of the set $\lbrace \ell(1),\ldots,\ell(n)\rbrace$ and define $\gamma_i:=\phi_{\ell(i)\ell}(c_i)\in A_{\ell}$ . Then \begin{eqnarray*} \phi_{j\ell}(a)&=& \phi_{j\ell}\big(\omega_j(\alpha_1,\ldots,\alpha_n)\big) \\ &=& \omega_{\ell}\big(\phi_{j\ell}(\alpha_1),\ldots,\phi_{j\ell}(\alpha_n)\big) \\ &=& \omega_{\ell}\big(\phi_{\ell(1)\ell}\circ \phi_{j\ell(1)}(\alpha_1),\ldots, \phi_{\ell(n)\ell}\circ \phi_{j\ell(n)}(\alpha_n)\big) \\ &=& \omega_{\ell}\big(\phi_{\ell(1)\ell}(c_1),\ldots,\phi_{\ell(n)\ell}(c_n)\big) \\ &=& \omega_{\ell}\big(\phi_{\ell(1)\ell}\circ \phi_{k\ell(1)}(\beta_1),\ldots, \phi_{\ell(n)\ell}\circ
\phi_{k\ell(n)}(\beta_n)\big) \\ &=& \omega_{\ell}\big(\phi_{k\ell}(\beta_1),\ldots,\phi_{k\ell}(\beta_n)\big) \\ &=& \phi_{k\ell}\big(\omega_k(\beta_1,\ldots,\beta_n)\big) \\ &=& \phi_{k\ell}(b), \end{eqnarray*}which shows that $a\sim b$ . 
Definition. Let $\mathcal{A}$ be a direct family of algebraic systems of the same type (say $O$ ) indexed by $I$ . The $O$ -algebra $A_{\infty}$ constructed above is called the direct limit of $\mathcal{A}$ . $A_{\infty}$ is alternatively written $\varinjlim A_i$ .
Remark. Dually, one can define an inverse family of algebraic systems, and its inverse limit. The inverse limit of an inverse family $\mathcal{A}$ is written $A^{\infty}$ or $\varprojlim A_i$ .
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