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:=. Since is directed and is finite, has an upper bound . Let . Define
Proposition 1.
is a well-defined -ary operation on .
Proof.
Suppose . Let be defined as above, and let . Similarly, are defined: , where . Let . We want to show that .
Since , . So there is . Let be the upper bound of the set and define . Then
which shows that . ∎
Definition. Let be a direct family of algebraic systems of the same type (say ) indexed by . The -algebra constructed above is called the direct limit of . is alternatively written .
Remark. Dually, one can define an inverse family of algebraic systems, and its inverse limit. The inverse limit of an inverse family is written or .
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 |