direct limit of algebraic systems

An immediate generalizationPlanetmathPlanetmath of the concept of the direct limitMathworldPlanetmath 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 homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. For completeness, we will spell out the definition in its entirety.

Let 𝒜={AiiI} be a family of algebraic systems of the same type (say, they are all O-algebrasMathworldPlanetmathPlanetmath), indexed by a non-empty set I. 𝒜 is said to be a direct family if

  1. 1.

    I is a directed setMathworldPlanetmath,

  2. 2.

    whenever ij in I, there is a homomorphism ϕij:AiAj,

  3. 3.

    ϕii is the identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on Ai,

  4. 4.

    if ijk, then ϕjkϕij=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 (iI). Take the disjoint unionMathworldPlanetmathPlanetmath of the underlying sets of each algebraic system, and call it A. Next, a binary relationMathworldPlanetmath is defined on A as follows:

given that aAi and bAj, ab iff there is Ak such that ϕik(a)=ϕjk(b).

It is shown here ( that is an equivalence relationMathworldPlanetmath 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 aA. 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 operationsMathworldPlanetmath ωi defined on Ai for all iI, we define an n-ary operation ω on A as follows:

for i=1,,n, pick aiAj(i), j(i)I. Let J:={j(i)i=1,,n}. Since I is directed and J is finite, J has an upper bound jI. Let αi=ϕj(i)j(ai). Define

Proposition 1.

ω is a well-defined n-ary operation on A.


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 biAk(i). Let b:=ωk(β1,,βn)Ak. We want to show that ab.

Since aibi, αiβi. So there is ci:=ϕj(i)(αi)=ϕk(i)(βi)A(i). Let be the upper bound of the set {(1),,(n)} and define γi:=ϕ(i)(ci)A. Then

ϕj(a) = ϕj(ωj(α1,,αn))
= ω(ϕj(α1),,ϕj(αn))
= ω(ϕ(1)ϕj(1)(α1),,ϕ(n)ϕj(n)(αn))
= ω(ϕ(1)(c1),,ϕ(n)(cn))
= ω(ϕ(1)ϕk(1)(β1),,ϕ(n)ϕk(n)(βn))
= ω(ϕk(β1),,ϕk(βn))
= ϕk(ωk(β1,,βn))
= ϕk(b),

which shows that ab. ∎

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 limAi.

Remark. Dually, one can define an inverse family of algebraic systems, and its inverse limitMathworldPlanetmath. The inverse limit of an inverse family 𝒜 is written A or limAi.

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