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Homesubdirect product of algebraic systems

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# subdirect product of algebraic systems

In this entry, all algebraic systems are of the same type. For each algebraic system, we drop the associated operator set for simplicity.

Let $A_{i}$ be algebraic systems indexed by $i\in I$. $B$ is called a *subdirect product* of $A_{i}$ if

1. $B$ is a subalgebra of the direct product of $A_{i}$.

2. for each $i\in I$, $\pi_{i}(B)=A_{i}$.

In the second condition, $\pi_{i}$ denotes the projection homomorphism $\prod A_{i}\to A_{i}$. By restriction, we may consider $\pi_{i}$ as homomorphisms $B\to A_{i}$. When $B$ is isomorphic to $\prod A_{i}$, then $B$ is a *trivial subdirect product* of $A_{i}$.

This generalizes the notion of a direct product, since in many instances, an algebraic system can not be decomposed into a direct product of algebras.

When all $A_{i}=C$ for some algebraic system $C$ of the same type, then $B$ is called a *subdirect power* of $C$.

Remarks.

1. 2. Let $B$ is a subdirect product of $A_{i}$, and $p_{i}:=(\pi_{i})_{B}$, the restriction of $\pi_{i}$ to $B$. Then $B/\ker(p_{i})\cong A_{i}$. In addition,

$\bigcap\{\ker(p_{i})\mid i\in I\}=\Delta,$ where $\Delta$ is the diagonal relation. To see the last equality, suppose $a,b\in B$ with $a\equiv b\;\;(\mathop{{\rm mod}}p_{i})$. Then $a(i)=\pi_{i}(a)=p_{i}(a)=p_{i}(b)=\pi_{i}(b)=b(i)$. Since this is true for every $i\in I$, $a=b$.

3. Conversely, if $A$ is an algebraic system and $\{\mathfrak{C}_{i}\mid i\in I\}$ is a set of congruences on $A$ such that

$\bigcap\{\mathfrak{C}_{i}\mid i\in I\}=\Delta.$ Then $A$ is isomorphic to a subdirect product of $A/\mathfrak{C}_{i}$.

4. An algebraic system is said to be

*subdirectly irreducible*if, whenever $\mathfrak{C}_{i}$ are congruences on $A$ and $\bigcap\{\mathfrak{C}_{i}\mid i\in I\}=\Delta$, then one of $\mathfrak{C}_{i}=\Delta$.5. Birkhoff’s Theorem on the Decomposition of an Algebraic System. Every algebraic system is isomorphic to a subdirect product of subdirectly irreducible algebraic systems. This works only when the algebraic system is finitary.

## Mathematics Subject Classification

08A62*no label found*08A05

*no label found*08B26

*no label found*

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