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 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 $J$ 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 $\prod A_i$.