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.
for each , .
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 .
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).
An algebraic system is said to be subdirectly irreducible if, whenever are congruences on and , then one of .
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|
|Date of creation||2013-03-22 16:44:51|
|Last modified on||2013-03-22 16:44:51|
|Last modified by||CWoo (3771)|
|Defines||trivial subdirect product|