PlanetMath: restricted direct product of algebraic systems

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restricted direct product of algebraic systems

(Definition)

Let $\lbrace A_i\mid i\in I\rbrace$ be a family of algebraic systems indexed by a set . Let be a Boolean ideal in , the Boolean algebra over the power set of . A subset of the direct product $\prod \lbrace A_i\mid i\in I\rbrace$ is called a restricted direct product of if

is a subalgebra of $\prod \lbrace A_i\mid i\in I\rbrace$ , and
given any $(a_i)\in B$ , we have that $(b_i)\in B$ iff $\lbrace i\in I\mid a_i\ne b_i\rbrace \in J$ .

If it is necessary to distinguish the different restricted direct products of

, we often specify the “restriction”, hence we say that

is a

-restricted direct product of

, or that

is restricted to

Here are some special restricted direct products:

If above, then is the direct product $\prod A_i$ , for if $(b_i)\in \prod A_i$ , then clearly $\lbrace i\in I\mid a_i\ne b_i\rbrace\in P(I)$ , where $(a_i)\in B$ ( is non-empty since it is a subalgebra). Therefore $(b_i)\in B$ .
This justifies calling the direct product the “unrestricted direct product” by some people.
If is the ideal consisting of all finite subsets of , then is called the weak direct product of .
If is the singleton $\lbrace \varnothing\rbrace$ , then is also a singleton: pick $a,b\in B$ , then $\lbrace i\mid a_i\ne b_i\rbrace = \varnothing$ , which is equivalent to saying that .

Remark. While the direct product of always exists, restricted direct products may not. For example, in the last case above, A $\varnothing$ -restricted direct product exists only when there is an element $a\in \prod A_i$ that is fixed by all operations on it: that is, if is an -ary operation on $\prod A_i$ , then $f(a,\ldots,a)=a$ . In this case, $\lbrace a\rbrace$ is a $\varnothing$ -restricted direct product of $\prod A_i$ .