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 $B/\ker (p_i)\cong A_i$. In addition,

   $$\bigcap \{\ker (p_i)\mid i\in I\}=\mathrm{\Delta },$$

   where $\mathrm{\Delta }$ is the diagonal relation. To see the last equality, suppose $a,b\in B$ with $a\equiv b(mod{p}_i)$. 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 $\{{ℭ}_i\mid i\in I\}$ is a set of congruences on $A$ such that

   $$\bigcap \{{ℭ}_i\mid i\in I\}=\mathrm{\Delta }.$$

   Then $A$ is isomorphic to a subdirect product of $A/{ℭ}_i$.

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

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	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