In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element. The free monoid on a set A is usually denoted A∗. The free semigroup on A is the subsemigroup of A∗ containing all elements except the empty string. It is usually denoted A+.[1][2]
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.[3]
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.
Free monoids (and monoids in general) are associative, by definition; that is, they are written without any parenthesis to show grouping or order of operation. The non-associative equivalent is the free magma.
- ^ Lothaire (1997, pp. 2–3), [1]
- ^ Pytheas Fogg (2002, p. 2)
- ^ Lothaire (1997, p. 5)