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⁺.