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 ϕij are now homomorphisms. For completeness, we will spell out the definition in its entirety.
Let 𝒜={Ai∣i∈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≤j in I, there is a homomorphism ϕij:Ai→Aj,
-
3.
ϕii is the identity
on Ai,
-
4.
if i≤j≤k, then ϕjk∘ϕij=ϕik.
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 Ai, indexed by I (i∈I). Take the disjoint union of the underlying sets of each algebraic system, and call it A. Next, a binary relation
∼ is defined on A as
follows:
given that a∈Ai and b∈Aj, a∼b iff there is Ak such that ϕik(a)=ϕjk(b).
It is shown here (http://planetmath.org/DirectLimitOfSets) that ∼ is an equivalence relation on A, so we can take the quotient A/∼, and denote it by A∞. Elements of A∞ are denoted by [a]I or [a] when there is no confusion, where a∈A. So A∞ is just the direct limit of Ai considered as sets.
Next, we want to turn A∞ into an O-algebra. Corresponding to each set of n-ary operations ωi defined on Ai for all i∈I, we define an n-ary operation ω on A∞ as follows:
for i=1,…,n, pick ai∈Aj(i), j(i)∈I. Let J:={j(i)∣i=1,…,n}. Since I is directed and J is finite, J has an upper bound j∈I. Let αi=ϕj(i)j(ai). Define
ω([a1],…,[an]):=[ωj(α1,…,αn)].
Proposition 1.
ω is a well-defined n-ary operation on A∞.
Proof.
Suppose [b1]=[a1],…,[bn]=[an]. Let αi be defined as above, and let a:=ωj(α1,…,αn)∈Aj. Similarly, βi are defined: βi:=ϕk(i)k(bi)∈Ak, where bi∈Ak(i). Let b:=ωk(β1,…,βn)∈Ak. We want to show that a∼b.
Since ai∼bi, αi∼βi. So there is ci:=ϕjℓ(i)(αi)=ϕkℓ(i)(βi)∈Aℓ(i). Let ℓ be the upper bound of the set {ℓ(1),…,ℓ(n)} and define γi:=ϕℓ(i)ℓ(ci)∈Aℓ. Then
ϕjℓ(a) | = | ϕjℓ(ωj(α1,…,αn)) | ||
= | ωℓ(ϕjℓ(α1),…,ϕjℓ(αn)) | |||
= | ωℓ(ϕℓ(1)ℓ∘ϕjℓ(1)(α1),…,ϕℓ(n)ℓ∘ϕjℓ(n)(αn)) | |||
= | ωℓ(ϕℓ(1)ℓ(c1),…,ϕℓ(n)ℓ(cn)) | |||
= | ωℓ(ϕℓ(1)ℓ∘ϕkℓ(1)(β1),…,ϕℓ(n)ℓ∘ϕkℓ(n)(βn)) | |||
= | ωℓ(ϕkℓ(β1),…,ϕkℓ(βn)) | |||
= | ϕkℓ(ωk(β1,…,βn)) | |||
= | ϕkℓ(b), |
which shows that a∼b. ∎
Definition. Let 𝒜 be a direct family of algebraic systems of the same type (say O) indexed by I. The O-algebra A∞ constructed above is called the direct limit of 𝒜. A∞ is alternatively written lim→Ai.
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∞ or lim←Ai.
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 |