direct limit of algebraic systems
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.
Direct Family of Algebraic Systems
The definition is almost identical to that of a direct family of sets, except that functions ${\varphi}_{ij}$ are now homomorphisms^{}. For completeness, we will spell out the definition in its entirety.
Let $\mathcal{A}=\{{A}_{i}\mid i\in I\}$ be a family of algebraic systems of the same type (say, they are all $O$algebras^{}), indexed by a nonempty set $I$. $\mathcal{A}$ is said to be a direct family if

1.
$I$ is a directed set^{},

2.
whenever $i\le j$ in $I$, there is a homomorphism ${\varphi}_{ij}:{A}_{i}\to {A}_{j}$,

3.
${\varphi}_{ii}$ is the identity^{} on ${A}_{i}$,

4.
if $i\le j\le k$, then ${\varphi}_{jk}\circ {\varphi}_{ij}={\varphi}_{ik}$.
An example of this is a direct family of sets. A homomorphism between two sets is just a function between the sets.
Direct Limit of Algebraic Systems
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 ${\varphi}_{ik}(a)={\varphi}_{jk}(b)$.
It is shown here (http://planetmath.org/DirectLimitOfSets) that $\sim $ is an equivalence relation^{} on $A$, so we can take the quotient $A/\sim $, and denote it by ${A}_{\mathrm{\infty}}$. Elements of ${A}_{\mathrm{\infty}}$ are denoted by ${[a]}_{I}$ or $[a]$ when there is no confusion, where $a\in A$. So ${A}_{\mathrm{\infty}}$ is just the direct limit of ${A}_{i}$ considered as sets.
Next, we want to turn ${A}_{\mathrm{\infty}}$ 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 ${A}_{\mathrm{\infty}}$ as follows:
for $i=1,\mathrm{\dots},n$, pick ${a}_{i}\in {A}_{j(i)}$, $j(i)\in I$. Let $J:=\{j(i)\mid i=1,\mathrm{\dots},n\}$. Since $I$ is directed and $J$ is finite, $J$ has an upper bound $j\in I$. Let ${\alpha}_{i}={\varphi}_{j(i)j}({a}_{i})$. Define
$$\omega ([{a}_{1}],\mathrm{\dots},[{a}_{n}]):=[{\omega}_{j}({\alpha}_{1},\mathrm{\dots},{\alpha}_{n})].$$
Proposition 1.
$\omega $ is a welldefined $n$ary operation on ${A}_{\mathrm{\infty}}$.
Proof.
Suppose $[{b}_{1}]=[{a}_{1}],\mathrm{\dots},[{b}_{n}]=[{a}_{n}]$. Let ${\alpha}_{i}$ be defined as above, and let $a:={\omega}_{j}({\alpha}_{1},\mathrm{\dots},{\alpha}_{n})\in {A}_{j}$. Similarly, ${\beta}_{i}$ are defined: ${\beta}_{i}:={\varphi}_{k(i)k}({b}_{i})\in {A}_{k}$, where ${b}_{i}\in {A}_{k(i)}$. Let $b:={\omega}_{k}({\beta}_{1},\mathrm{\dots},{\beta}_{n})\in {A}_{k}$. We want to show that $a\sim b$.
Since ${a}_{i}\sim {b}_{i}$, ${\alpha}_{i}\sim {\beta}_{i}$. So there is ${c}_{i}:={\varphi}_{j\mathrm{\ell}(i)}({\alpha}_{i})={\varphi}_{k\mathrm{\ell}(i)}({\beta}_{i})\in {A}_{\mathrm{\ell}(i)}$. Let $\mathrm{\ell}$ be the upper bound of the set $\{\mathrm{\ell}(1),\mathrm{\dots},\mathrm{\ell}(n)\}$ and define ${\gamma}_{i}:={\varphi}_{\mathrm{\ell}(i)\mathrm{\ell}}({c}_{i})\in {A}_{\mathrm{\ell}}$. Then
${\varphi}_{j\mathrm{\ell}}(a)$  $=$  ${\varphi}_{j\mathrm{\ell}}\left({\omega}_{j}({\alpha}_{1},\mathrm{\dots},{\alpha}_{n})\right)$  
$=$  ${\omega}_{\mathrm{\ell}}({\varphi}_{j\mathrm{\ell}}({\alpha}_{1}),\mathrm{\dots},{\varphi}_{j\mathrm{\ell}}({\alpha}_{n}))$  
$=$  ${\omega}_{\mathrm{\ell}}({\varphi}_{\mathrm{\ell}(1)\mathrm{\ell}}\circ {\varphi}_{j\mathrm{\ell}(1)}({\alpha}_{1}),\mathrm{\dots},{\varphi}_{\mathrm{\ell}(n)\mathrm{\ell}}\circ {\varphi}_{j\mathrm{\ell}(n)}({\alpha}_{n}))$  
$=$  ${\omega}_{\mathrm{\ell}}({\varphi}_{\mathrm{\ell}(1)\mathrm{\ell}}({c}_{1}),\mathrm{\dots},{\varphi}_{\mathrm{\ell}(n)\mathrm{\ell}}({c}_{n}))$  
$=$  ${\omega}_{\mathrm{\ell}}({\varphi}_{\mathrm{\ell}(1)\mathrm{\ell}}\circ {\varphi}_{k\mathrm{\ell}(1)}({\beta}_{1}),\mathrm{\dots},{\varphi}_{\mathrm{\ell}(n)\mathrm{\ell}}\circ {\varphi}_{k\mathrm{\ell}(n)}({\beta}_{n}))$  
$=$  ${\omega}_{\mathrm{\ell}}({\varphi}_{k\mathrm{\ell}}({\beta}_{1}),\mathrm{\dots},{\varphi}_{k\mathrm{\ell}}({\beta}_{n}))$  
$=$  ${\varphi}_{k\mathrm{\ell}}\left({\omega}_{k}({\beta}_{1},\mathrm{\dots},{\beta}_{n})\right)$  
$=$  ${\varphi}_{k\mathrm{\ell}}(b),$ 
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 ${A}_{\mathrm{\infty}}$ constructed above is called the direct limit of $\mathcal{A}$. ${A}_{\mathrm{\infty}}$ is alternatively written $\underrightarrow{\mathrm{lim}}{A}_{i}$.
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 ${A}^{\mathrm{\infty}}$ or $\underleftarrow{\mathrm{lim}}{A}_{i}$.
Title  direct limit of algebraic systems 
Canonical name  DirectLimitOfAlgebraicSystems 
Date of creation  20130322 16:53:56 
Last modified on  20130322 16:53:56 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 08B25 
Synonym  direct system of algebraic systems 
Synonym  inverse system of algebraic systems 
Synonym  projective system of algebraic systems 
Related topic  DirectLimitOfSets 
Defines  direct family of algebraic systems 
Defines  inverse family of algebraic systems 
Defines  inverse limit of algebraic systems 