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 Ai be algebraic systems indexed by iI. B is called a subdirect productPlanetmathPlanetmath of Ai if

  1. 1.

    B is a subalgebraPlanetmathPlanetmathPlanetmath of the direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of Ai.

  2. 2.

    for each iI, πi(B)=Ai.

In the second condition, πi denotes the projection homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath AiAi. By restrictionPlanetmathPlanetmath, we may consider πi as homomorphisms BAi. When B is isomorphic to Ai, then B is a trivial subdirect product of Ai.

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 Ai=C for some algebraic system C of the same type, then B is called a subdirect power of C.


  1. 1.

    A very simple example of a subdirect product is the following: let A1=A2={1,2,3}. Then the subset B={(x,y)A1×A2xy} is a subdirect product of the sets A1 and A2 (considered as algebraic systems with no operators).

  2. 2.

    Let B is a subdirect product of Ai, and pi:=(πi)B, the restriction of πi to B. Then B/ker(pi)Ai. In additionPlanetmathPlanetmath,


    where Δ is the diagonal relation. To see the last equality, suppose a,bB with ab(modpi). Then a(i)=πi(a)=pi(a)=pi(b)=πi(b)=b(i). Since this is true for every iI, a=b.

  3. 3.

    Conversely, if A is an algebraic system and {iiI} is a set of congruencesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on A such that


    Then A is isomorphic to a subdirect product of A/i.

  4. 4.

    An algebraic system is said to be subdirectly irreducible if, whenever i are congruences on A and {iiI}=Δ, then one of i=Δ.

  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