direct limit of algebraic systems
Direct Family of Algebraic Systems
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
given that and , iff there is such that .
It is shown here (http://planetmath.org/DirectLimitOfSets) that is an equivalence relation on , so we can take the quotient , and denote it by . Elements of are denoted by or when there is no confusion, where . So is just the direct limit of considered as sets.
Next, we want to turn into an -algebra. Corresponding to each set of -ary operations defined on for all , we define an -ary operation on as follows:
for , pick , . Let . Since is directed and is finite, has an upper bound . Let . Define
is a well-defined -ary operation on .
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|
|Date of creation||2013-03-22 16:53:56|
|Last modified on||2013-03-22 16:53:56|
|Last modified by||CWoo (3771)|
|Synonym||direct system of algebraic systems|
|Synonym||inverse system of algebraic systems|
|Synonym||projective system of algebraic systems|
|Defines||direct family of algebraic systems|
|Defines||inverse family of algebraic systems|
|Defines||inverse limit of algebraic systems|