direct product of algebras

In this entry, let O be a fixed operator set. All algebraic systems have the same type (they are all O-algebrasMathworldPlanetmathPlanetmath).

Let {AiiI} be a set of algebraic systems of the same type (O) indexed by I. Let us form the Cartesian productMathworldPlanetmath of the underlying sets and call it A:


Recall that element a of A is a function from I to Ai such that for each iI, a(i)Ai.

For each ωO with arity n, let ωAi be the corresponding n-ary operator on Ai. Define ωA:AnA by

ωA(a1,,an)(i)=ωAi(a1(i),,an(i)) for all iI.

One readily checks that ωA is a well-defined n-ary operator on A. A equipped with all ωA on A is an O-algebra, and is called the direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of Ai. Each Ai is called a direct factor of A.

If each Ai=B, where B is an O-algebra, then we call A the direct power of B and we write A as BI (keep in mind the isomorphicPlanetmathPlanetmathPlanetmath identifications).

If A is the direct product of Ai, then for each iI we can associate a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath πi:AAi called a projection given by πi(a)=a(i). It is a homomorphism because πi(ωA(a1,,an))=ωA(a1,,an)(i)=ωAi(a1(i),,an(i))=ωAi(πi(a1),,πi(an)).

Remark. The direct product of a single algebraic system is the algebraic system itself. An empty direct product is defined to be a trivial algebraic system (one-element algebra).

Title direct product of algebras
Canonical name DirectProductOfAlgebras
Date of creation 2013-03-22 16:44:35
Last modified on 2013-03-22 16:44:35
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 08A05
Classification msc 08A62
Defines direct product
Defines direct factor
Defines direct power
Defines projection
Defines empty direct product