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[parent] subdirect product of algebraic systems (Definition)

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=\lbrace 1,2,3\rbrace$. Then the subset $ B=\lbrace (x,y)\in A_1\times A_2 \mid x\le y \rbrace$ 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,
    $\displaystyle \bigcap \lbrace \ker(p_i)\mid i\in I\rbrace=\Delta,$
    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. Conversely, if $ A$ is an algebraic system and $ \lbrace \mathfrak{C}_i\mid i\in I\rbrace$ is a set of congruences on $ A$ such that
    $\displaystyle \bigcap \lbrace \mathfrak{C}_i\mid i\in I\rbrace=\Delta.$
    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 \lbrace \mathfrak{C}_i\mid i\in I\rbrace=\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.



"subdirect product of algebraic systems" is owned by CWoo. [ full author list (2) ]
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Also defines:  subdirect product, subdirect power, subdirectly irreducible, trivial subdirect product

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Attachments:
subdirect product of rings (Definition) by CWoo
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Cross-references: decomposition, congruences, equality, diagonal relation, addition, operators, subset, simple, direct product of algebras, isomorphic, restriction, homomorphism, projection, direct product, subalgebra, indexed by, operator set, type, algebraic systems
There are 4 references to this entry.

This is version 7 of subdirect product of algebraic systems, born on 2007-02-24, modified 2007-08-05.
Object id is 8972, canonical name is SubdirectProductOfAlgebraicSystems.
Accessed 1621 times total.

Classification:
AMS MSC: 08A62 (General algebraic systems :: Algebraic structures :: Finitary algebras)
 08A05 (General algebraic systems :: Algebraic structures :: Structure theory)
 08B26 (General algebraic systems :: Varieties :: Subdirect products and subdirect irreducibility)
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