structural:scholar_edition, structural:reformatting
Examples

The creation of free objects proceeds in two steps. For algebras that conform to the associative law, the first step is to consider the collection of all possible words formed from an alphabet. Then one imposes a set of equivalence relations upon the words, where the relations are the defining relations of the algebraic object at hand. The free object then consists of the set of equivalence classes.

Consider, for example, the construction of the free group in two generators. One starts with an alphabet consisting of the five letters { e, a, b, a − 1, b − 1 } {\displaystyle \{e,a,b,a^{-1},b^{-1}\}}. In the first step, there is not yet any assigned meaning to the "letters" a − 1 {\displaystyle a^{-1}} or b − 1 {\displaystyle b^{-1}}; these will be given later, in the second step. Thus, one could equally well start with the alphabet in five letters that is S = { a, b, c, d, e } {\displaystyle S=\{a,b,c,d,e\}}. In this example, the set of all words or strings W (S) {\displaystyle W(S)} will include strings such as aebecede and abdc, and so on, of arbitrary finite length, with the letters arranged in every possible order.

In the next step, one imposes a set of equivalence relations. The equivalence relations for a group are that of multiplication by the identity, g e = e g = g {\displaystyle ge=eg=g}, and the multiplication of inverses: g g − 1 = g − 1 g = e {\displaystyle gg^{-1}=g^{-1}g=e}. Applying these relations to the strings above, one obtains

where it was understood that c {\displaystyle c} is a stand-in for a − 1 {\displaystyle a^{-1}}, and d {\displaystyle d} is a stand-in for b − 1 {\displaystyle b^{-1}}, while e {\displaystyle e} is the identity element. Similarly, one has

Denoting the equivalence relation or congruence by ∼ {\displaystyle \sim }, the free object is then the collection of equivalence classes of words. Thus, in this example, the free group in two generators is the quotient

This is often written as F 2 = W (S) / E {\displaystyle F_{2}=W(S)/E} where W (S) = { a 1 a 2 … a n | a k ∈ S; n ∈ N } {\displaystyle W(S)=\{a_{1}a_{2}\ldots a_{n}\,\vert \;a_{k}\in S\,;\,n\in \mathbb {N} \}} is the set of all words, and E = { a 1 a 2 … a n | e = a 1 a 2 … a n; a k ∈ S; n ∈ N } {\displaystyle E=\{a_{1}a_{2}\ldots a_{n}\,\vert \;e=a_{1}a_{2}\ldots a_{n}\,;\,a_{k}\in S\,;\,n\in \mathbb {N} \}} is the equivalence class of the identity, after the relations defining a group are imposed.

A simpler example are the free monoids. The free monoid on a set X, is the monoid of all finite strings using X as alphabet, with operation concatenation of strings. The identity is the empty string. In essence, the free monoid is simply the set of all words, with no equivalence relations imposed. This example is developed further in the article on the Kleene star.