Free Monoid

In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from A. It is usually denoted A∗. The identity element is the unique sequence of zero elements, often called the empty string and denoted by ε or λ, and the monoid operation is string concatenation. The free semigroup on A is the subsemigroup of A∗ containing all elements except the empty string. It is usually denoted A+.

More generally, an abstract monoid (or semigroup) S is described as free if it is isomorphic to the free monoid (or semigroup) on some set.

As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective categories of monoids and semigroups. It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup). The study of semigroups as images of free semigroups is called combinatorial semigroup theory.