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restricted direct product of algebraic systems
Let $\{A_{i}\mid i\in I\}$ be a family of algebraic systems indexed by a set $I$. Let $J$ be a Boolean ideal in $P(I)$, the Boolean algebra over the power set of $I$. A subset $B$ of the direct product $\prod\{A_{i}\mid i\in I\}$ is called a restricted direct product of $A_{i}$ if
1. $B$ is a subalgebra of $\prod\{A_{i}\mid i\in I\}$, and
2. given any $(a_{i})\in B$, we have that $(b_{i})\in B$ iff $\{i\in I\mid a_{i}\neq b_{i}\}\in J$.
If it is necessary to distinguish the different restricted direct products of $A_{i}$, we often specify the “restriction”, hence we say that $B$ is a $J$restricted direct product of $A_{i}$, 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 $\prod A_{i}$, for if $(b_{i})\in\prod A_{i}$, then clearly $\{i\in I\mid a_{i}\neq b_{i}\}\in P(I)$, where $(a_{i})\in B$ ($B$ is nonempty since it is a subalgebra). Therefore $(b_{i})\in B$.
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 $A_{i}$.

If $J$ is the singleton $\{\varnothing\}$, then $B$ is also a singleton: pick $a,b\in B$, then $\{i\mid a_{i}\neq b_{i}\}=\varnothing$, which is equivalent to saying that $(a_{i})=(b_{i})$.
Remark. While the direct product of $A_{i}$ 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 $f$ is an $n$ary operation on $\prod A_{i}$, then $f(a,\ldots,a)=a$. In this case, $\{a\}$ is a $\varnothing$restricted direct product of $\prod A_{i}$.
References
 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
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