free group in two generators. One starts with an alphabet consisting of the five letters . In the first step, there is not yet any assigned meaning to the "letters" or ; these will be given later, in the second step. Thus, one could equally well start with the alphabet in five letters that is . In this example, the set of all words or strings 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, , and the multiplication of inverses: . Applying these relations to the strings above, one obtains where it was understood that is a stand-in for , and is a stand-in for , while is the identity element. Similarly, one has Denoting the equivalence relation or congruence by , 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 where is the set of all words, and 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. General case In the general case, the algebraic relations need not be associative, in which case the starting point is not the set of all words, but rather, strings punctuated with parentheses, which are used to indicate the non-associative groupings of letters. Such a string may equivalently be represented by a binary tree or a free magma; the leaves of the tree are the letters from the alphabet. The algebraic relations may then be general arities or finitary relations on the leaves of the tree. Rather than starting with the collection of all possible parenthesized strings, it can be more convenient to start with the Herbrand universe. Properly describing or enumerating the contents of a free object can be easy or