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 be algebraic systems indexed by . is called a subdirect product of if
-
1.
is a subalgebra of the direct product of .
-
2.
for each , .
In the second condition, denotes the projection homomorphism . By restriction, we may consider as homomorphisms . When is isomorphic to , then is a trivial subdirect product of .
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 for some algebraic system of the same type, then is called a subdirect power of .
Remarks.
-
1.
A very simple example of a subdirect product is the following: let . Then the subset is a subdirect product of the sets and (considered as algebraic systems with no operators).
-
2.
Let is a subdirect product of , and , the restriction of to . Then . In addition,
where is the diagonal relation. To see the last equality, suppose with . Then . Since this is true for every , .
-
3.
Conversely, if is an algebraic system and is a set of congruences on such that
Then is isomorphic to a subdirect product of .
-
4.
An algebraic system is said to be subdirectly irreducible if, whenever are congruences on and , then one of .
-
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 |