free algebra
Let be a class of algebraic systems (of the same type ). Consider an algebra generated by (http://planetmath.org/SubalgebraOfAnAlgebraicSystem) a set indexed by . is said to be a free algebra over , with free generating set , if for any algebra with any subset , there is a homomorphism such that .
If we define to be and to be , then freeness of means the existence of such that .
Note that above is necessarily unique, since generates . For any -ary polynomial over , any , .
For example, any free group is a free algebra in the class of groups. In general, however, free algebras do not always exist in an arbitrary class of algebras.
Remarks.
-
•
is free over itself (meaning consists of only) iff is free over some equational class.
-
•
If is an equational class, then free algebras exist in .
-
•
Any term algebra of a given structure over some set of variables is a free algebra with free generating set .
Title | free algebra |
---|---|
Canonical name | FreeAlgebra |
Date of creation | 2013-03-22 16:51:05 |
Last modified on | 2013-03-22 16:51:05 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 6 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 08B20 |
Synonym | free algebraic system |
Related topic | TermAlgebra |
Defines | free generating set |