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Homedirect limit of algebraic systems

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# 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 $\phi_{{ij}}$ 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 non-empty set $I$. $\mathcal{A}$ is said to be a *direct family* if

1. $I$ is a directed set,

2. whenever $i\leq j$ in $I$, there is a homomorphism $\phi_{{ij}}:A_{i}\to A_{j}$,

3. $\phi_{{ii}}$ is the identity on $A_{i}$,

4. if $i\leq j\leq k$, 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.

# 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 $\phi_{{ik}}(a)=\phi_{{jk}}(b)$.

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 $J:=\{j(i)\mid i=1,\ldots,n\}$. 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 $\{\ell(1),\ldots,\ell(n)\}$ and define $\gamma_{i}:=\phi_{{\ell(i)\ell}}(c_{i})\in A_{{\ell}}$. Then

$\displaystyle\phi_{{j\ell}}(a)$ | $\displaystyle=$ | $\displaystyle\phi_{{j\ell}}\big(\omega_{j}(\alpha_{1},\ldots,\alpha_{n})\big)$ | ||

$\displaystyle=$ | $\displaystyle\omega_{{\ell}}\big(\phi_{{j\ell}}(\alpha_{1}),\ldots,\phi_{{j% \ell}}(\alpha_{n})\big)$ | |||

$\displaystyle=$ | $\displaystyle\omega_{{\ell}}\big(\phi_{{\ell(1)\ell}}\circ\phi_{{j\ell(1)}}(% \alpha_{1}),\ldots,\phi_{{\ell(n)\ell}}\circ\phi_{{j\ell(n)}}(\alpha_{n})\big)$ | |||

$\displaystyle=$ | $\displaystyle\omega_{{\ell}}\big(\phi_{{\ell(1)\ell}}(c_{1}),\ldots,\phi_{{% \ell(n)\ell}}(c_{n})\big)$ | |||

$\displaystyle=$ | $\displaystyle\omega_{{\ell}}\big(\phi_{{\ell(1)\ell}}\circ\phi_{{k\ell(1)}}(% \beta_{1}),\ldots,\phi_{{\ell(n)\ell}}\circ\phi_{{k\ell(n)}}(\beta_{n})\big)$ | |||

$\displaystyle=$ | $\displaystyle\omega_{{\ell}}\big(\phi_{{k\ell}}(\beta_{1}),\ldots,\phi_{{k\ell% }}(\beta_{n})\big)$ | |||

$\displaystyle=$ | $\displaystyle\phi_{{k\ell}}\big(\omega_{k}(\beta_{1},\ldots,\beta_{n})\big)$ | |||

$\displaystyle=$ | $\displaystyle\phi_{{k\ell}}(b),$ |

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 $A_{{\infty}}$ constructed above is called the *direct limit* of $\mathcal{A}$. $A_{{\infty}}$ is alternatively written $\underrightarrow{\lim}A_{i}$.

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 $A^{{\infty}}$ or $\underleftarrow{\lim}A_{i}$.

## Mathematics Subject Classification

08B25*no label found*

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