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 $𝒜=\{A_i\mid i\in I\}$ be a family of algebraic systems of the same type (say, they are all $O$-algebras), indexed by a non-empty set $I$. $𝒜$ 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 $𝒜$ 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 well-defined $n$-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 $𝒜$ 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 $𝒜$. $A_{\mathrm{\infty }}$ 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 $𝒜$ is written $A^{\mathrm{\infty }}$ or $\underleftarrow{\lim }{A}_i$.

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