free semigroup (original) (raw)

Let X be a set. We define the power of X in a language-theoretical manner as

Xn={x1⁢x2⁢…⁢xn∣xj∈X⁢ for all ⁢j∈{1,…,n}}

for all n∈ℕ∖{0}, and

where ε∉X. Note that the set X is not necessarily an alphabet, that is, it may be infinite ; for example, we may choose X=ℝ.

We define the sets X+ and X* as

and

The elements of X* are called words on X, and ε is called the empty word on X.

References

1 J.M. Howie,Fundamentals of Semigroup Theory, Oxford University Press, Oxford, 1991.