# 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. 1.

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

2. 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. 1.

A very simple example of a subdirect product is the following: let $A_{1}=A_{2}=\{1,2,3\}$. Then the subset $B=\{(x,y)\in A_{1}\times A_{2}\mid x\leq y\}$ is a subdirect product of the sets $A_{1}$ and $A_{2}$ (considered as algebraic systems with no operators).

2. 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 $a\equiv b\pmod{p_{i}}$. 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. 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. 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. 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.

 Title subdirect product of algebraic systems Canonical name SubdirectProductOfAlgebraicSystems Date of creation 2013-03-22 16:44:51 Last modified on 2013-03-22 16:44:51 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 10 Author CWoo (3771) Entry type Definition Classification msc 08A62 Classification msc 08A05 Classification msc 08B26 Defines subdirect product Defines subdirect power Defines subdirectly irreducible Defines trivial subdirect product