restricted direct product of algebraic systems

Let {AiiI} be a family of algebraic systems indexed by a set I. Let J be a Boolean ideal in P(I), the Boolean algebraMathworldPlanetmath over the power setMathworldPlanetmath of I. A subset B of the direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath {AiiI} is called a restricted direct product of Ai if

  1. 1.

    B is a subalgebraPlanetmathPlanetmathPlanetmath of {AiiI}, and

  2. 2.

    given any (ai)B, we have that (bi)B iff {iIaibi}J.

If it is necessary to distinguish the different restricted direct products of Ai, we often specify the “restrictionPlanetmathPlanetmath”, hence we say that B is a J-restricted direct product of Ai, or that B is restricted to J.

Here are some special restricted direct products:

  • If J=P(I) above, then B is the direct product Ai, for if (bi)Ai, then clearly {iIaibi}P(I), where (ai)B (B is non-empty since it is a subalgebra). Therefore (bi)B.

    This justifies calling the direct product the “unrestricted direct product” by some people.

  • If J is the ideal consisting of all finite subsets of I, then B is called the weak direct product of Ai.

  • If J is the singleton {}, then B is also a singleton: pick a,bB, then {iaibi}=, which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to saying that (ai)=(bi).

Remark. While the direct product of Ai always exists, restricted direct products may not. For example, in the last case above, A -restricted direct product exists only when there is an element aAi that is fixed by all operationsMathworldPlanetmath on it: that is, if f is an n-ary operation on Ai, then f(a,,a)=a. In this case, {a} is a -restricted direct product of Ai.


Title restricted direct product of algebraic systems
Canonical name RestrictedDirectProductOfAlgebraicSystems
Date of creation 2013-03-22 17:05:57
Last modified on 2013-03-22 17:05:57
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 08B25
Defines restricted direct product
Defines weak direct product