# 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. 1.

$I$ is a directed set,

2. 2.

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

3. 3.

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

4. 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 (http://planetmath.org/DirectLimitOfSets) that $\sim$ is an equivalence relation on $A$, so we can take the quotient $A/\sim$, and denote it by $A_{\infty}$. 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 $\varinjlim A_{i}$.

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 $\varprojlim A_{i}$.

 Title direct limit of algebraic systems Canonical name DirectLimitOfAlgebraicSystems Date of creation 2013-03-22 16:53:56 Last modified on 2013-03-22 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