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 $\{i\in I\mid {a}_{i}\ne {b}_{i}\}\in J$.
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 $\{i\in I\mid {a}_{i}\ne {b}_{i}\}\in P(I)$, 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 $\{\mathrm{\varnothing}\}$, then $B$ is also a singleton: pick $a,b\in B$, then $\{i\mid {a}_{i}\ne {b}_{i}\}=\mathrm{\varnothing}$, 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 $\mathrm{\varnothing}$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 $f(a,\mathrm{\dots},a)=a$. In this case, $\{a\}$ is a $\mathrm{\varnothing}$restricted direct product of $\prod {A}_{i}$.
References
 1 G. Grätzer: Universal Algebra^{}, 2nd Edition, Springer, New York (1978).
Title  restricted direct product of algebraic systems 

Canonical name  RestrictedDirectProductOfAlgebraicSystems 
Date of creation  20130322 17:05:57 
Last modified on  20130322 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 