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

2.
Let $B$ is a subdirect product of ${A}_{i}$, and ${p}_{i}:={({\pi}_{i})}_{B}$, the restriction of ${\pi}_{i}$ to $B$. Then $B/\mathrm{ker}({p}_{i})\cong {A}_{i}$. In addition^{},
$$\bigcap \{\mathrm{ker}({p}_{i})\mid i\in I\}=\mathrm{\Delta},$$ where $\mathrm{\Delta}$ is the diagonal relation. To see the last equality, suppose $a,b\in B$ with $a\equiv b\phantom{\rule{veryverythickmathspace}{0ex}}(mod{p}_{i})$. 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 $\{{\u212d}_{i}\mid i\in I\}$ is a set of congruences^{} on $A$ such that
$$\bigcap \{{\u212d}_{i}\mid i\in I\}=\mathrm{\Delta}.$$ Then $A$ is isomorphic to a subdirect product of $A/{\u212d}_{i}$.

4.
An algebraic system is said to be subdirectly irreducible if, whenever ${\u212d}_{i}$ are congruences on $A$ and $\bigcap \{{\u212d}_{i}\mid i\in I\}=\mathrm{\Delta}$, then one of ${\u212d}_{i}=\mathrm{\Delta}$.

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.
Title  subdirect product of algebraic systems 
Canonical name  SubdirectProductOfAlgebraicSystems 
Date of creation  20130322 16:44:51 
Last modified on  20130322 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 