free algebra

Let 𝒦 be a class of algebraic systems (of the same type τ). Consider an algebraMathworldPlanetmathPlanetmathPlanetmath A𝒦 generated by ( a set X={xi} indexed by iI. A is said to be a free algebraMathworldPlanetmath over 𝒦, with free generating set X, if for any algebra B𝒦 with any subset {yiiI}B, there is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ϕ:AB such that ϕ(xi)=yi.

If we define f:IA to be f(i)=xi and g:IB to be g(i)=yi, then freeness of A means the existence of ϕ:AB such that ϕf=g.

Note that ϕ above is necessarily unique, since {xi} generates A. For any n-ary polynomialMathworldPlanetmathPlanetmath p over A, any z1,,zn{xiiI}, ϕ(p(z1,,zn))=p(ϕ(z1),,ϕ(zn)).

For example, any free groupMathworldPlanetmath is a free algebra in the class of groups. In general, however, free algebras do not always exist in an arbitrary class of algebras.


  • A is free over itself (meaning 𝒦 consists of A only) iff A is free over some equational class.

  • If 𝒦 is an equational class, then free algebras exist in 𝒦.

  • Any term algebra of a given structureMathworldPlanetmath τ over some set X of variables is a free algebra with free generating set X.

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