# 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. 1.

$B$ is a subalgebra of $\prod\{A_{i}\mid i\in I\}$, and

2. 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 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 $\{\varnothing\}$, 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: , 2nd Edition, Springer, New York (1978).
Title restricted direct product of algebraic systems RestrictedDirectProductOfAlgebraicSystems 2013-03-22 17:05:57 2013-03-22 17:05:57 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 08B25 restricted direct product weak direct product