# reduced direct product

Let $\{A_{i}\mid i\in I\}$ be a set of algebraic systems of the same type, indexed by $I$. Let $A$ be the direct product of the $A_{i}$’s. For any $a,b\in A$, set

 $\operatorname{supp}(a,b):=\{k\in I\mid a(k)\neq b(k)\}.$

Consider a Boolean ideal $L$ of the Boolean algebra $P(I)$ of $I$. Define a binary relation $\Theta_{L}$ on $A$ as follows:

 $(a,b)\in\Theta_{L}\quad\mbox{ iff }\quad\operatorname{supp}(a,b)\in L.$
###### Lemma 1.

$\Theta_{L}$ defined above is a congruence relation on $A$.

###### Proof.

Since $L$ is an ideal $\varnothing\in L$. Therefore, $(a,a)\in\Theta_{L}$, since $\{k\in I\mid a(k)\neq a(k)\}=\varnothing$. Clearly, $\Theta_{L}$ is symmetric. For transitivity, suppose $(a,b,(b,c)\in\Theta_{L}$. If $a(k)\neq c(k)$ for some $k\in I$, then either $a(k)\neq b(k)$ or $b(k)\neq c(k)$ (a contrapositive argument). So

 $\operatorname{supp}(a,c)\subseteq\operatorname{supp}(a,b)\cup\operatorname{% supp}(b,c).$

Since $L$ is an ideal, $\operatorname{supp}(a,c)\in L$, so $(a,c)\in\Theta_{L}$, and $\Theta_{L}$ is an equivalence relation on $A$.

Next, let $\omega$ be an $n$-ary operator on $A$ and $a_{j}\equiv b_{j}\pmod{\Theta_{L}}$, where $j=1,\ldots,n$. We want to show that $\omega(a_{1},\ldots,a_{n})\equiv\omega(b_{1},\ldots,b_{n})\pmod{\Theta_{L}}$. Let $\omega_{i}$ be the associated $n$-ary operators on $A_{i}$. If $\omega(a_{1},\ldots,a_{n})(k)\neq\omega(b_{1},\ldots,b_{n})(k)$, then $\omega_{k}(a_{1}(k),\ldots,a_{n}(k))\neq\omega_{k}(b_{1}(k),\ldots,b_{n}(k))$, which implies that $a_{j}(k)\neq b_{j}(k)$ for some $j=1,\ldots,n$. This implies that

 $\operatorname{supp}(\omega(a_{1},\ldots,a_{n}),\omega(b_{1},\ldots,b_{n}))% \subseteq\bigcup_{j=1}^{n}\operatorname{supp}(a_{j},b_{j}).$

Since $L$ is an ideal, and each $\operatorname{supp}(a_{j},b_{j})\in L$, we have that $\operatorname{supp}(\omega(a_{1},\ldots,a_{n}),\omega(b_{1},\ldots,b_{n}))\in L$ as well, this means that $\omega(a_{1},\ldots,a_{n})\equiv\omega(b_{1},\ldots,b_{n})\pmod{\Theta_{L}}$. ∎

Definition. Let $A=\prod\{A_{i}\mid i\in I\}$, $L$ be a Boolean ideal of $P(I)$ and $\Theta_{L}$ be defined as above. The quotient algebra $A/\Theta_{L}$ is called the $L$-reduced direct product of $A_{i}$. The $L$-reduced direct product of $A_{i}$ is denoted by $\prod_{L}\{A_{i}\mid i\in I\}$. Given any element $a\in A$, its image in the reduced direct product $\prod_{L}\{A_{i}\mid i\in I\}$ is given by $[a]\Theta_{L}$, or $[a]$ for short.

Example. Let $A=A_{1}\times\cdots\times A_{n}$, and let $L$ be the principal ideal generated by $1$. Then $L=\{\varnothing,\{1\}\}$. The congruence $\Theta_{L}$ is given by $(a_{1},\ldots,a_{n})\equiv(b_{1},\ldots,b_{n})\pmod{\Theta_{L}}$ iff $\{i\mid a_{i}\neq b_{i}\}=\varnothing$ or $\{1\}$. This implies that $a_{i}=b_{i}$ for all $i=2,\ldots,n$. In other words, $\Theta_{L}$ is isomorphic to the direct product of $A_{2}\times\cdots\times A_{n}$. Therefore, the $L$-reduced direct product of $A_{i}$ is isomorphic to $A_{1}$.

The example above can be generalized: if $J\subseteq I$, then

 $\prod{}_{P(J)}\{A_{i}\mid i\in I\}\cong\prod\{A_{i}\mid i\in I-J\}.$

For $a\in A=\prod\{A_{i}\mid i\in I\}$, write $a=(a_{i})_{i\in I}$. It is not hard to see that the map $f:\prod_{P(J)}\{A_{i}\mid i\in I\}\to\prod\{A_{i}\mid i\in I-J\}$ given by $f([a])=(a_{i})_{i\in I-J}$ is the required isomorphism.

Remark. The definition of a reduced direct product in terms of a Boolean ideal can be equivalently stated in terms of a Boolean filter $F$. All there is to do is to replace $\operatorname{supp}(a,b)$ by its complement: $\operatorname{supp}(a,b)^{c}:=\{k\in I\mid a(k)=b(k)\}$. The congruence relation is now $\Theta_{F^{\prime}}$, where $F^{\prime}=\{I-J\mid J\in F\}$ is the ideal complement of $F$. When $F$ is prime, the $F^{\prime}$-reduced direct product is called a prime product, or an ultraproduct, since any prime filter is also called an ultrafilter. Ultraproducts can be more generally defined over arbitrary structures.

## References

• 1 G. Grätzer: , 2nd Edition, Springer, New York (1978).
Title reduced direct product ReducedDirectProduct 2013-03-22 17:10:11 2013-03-22 17:10:11 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 08B25 ultraproduct prime product