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

for all $n\in\mathbb{N}\setminus\left\{0\right\}$, and

where $\varepsilon\notin X$. Note that the set $X$ is not necessarily an alphabet, that is, it may be infinite; for example, we may choose $X=\mathbb{R}$.

We define the sets $X^{+}$ and $X^{*}$ as

and

The elements of $X^{*}$ are called words on $X$, and $\varepsilon$ is called the empty word on $X$.

We define the juxtaposition of two words $v,w\in X^{*}$ as

where $v=v_{1}v_{2}\ldots v_{n}$ and $w=w_{1}w_{2}\ldots w_{m}$, with $v_{i},w_{j}\in X$ for each $i$ and $j$. It is easy to see that the juxtaposition is associative, so if we equip $X^{+}$ and $X^{*}$ with it we obtain respectively a semigroup and a monoid. Moreover, $X^{+}$ is the free semigroup on $X$ and $X^{*}$ is the free monoid on $X$, in the sense that they solve the following universal mapping problem: given a semigroup $S$ (resp. a monoid $M$) and a map $\Phi\colon X\to S$ (resp. $\Phi\colon X\to M$), a semigroup homomorphism $\overline{\Phi}\colon X^{+}\to S$ (resp. a monoid homomorphism $\overline{\Phi}\colon X^{*}\to M$) exists such that the following diagram commutes:

(resp.

), where $\iota\colon X\to X^{+}$ (resp. $\iota\colon X\to X^{*}$) is the inclusion map. It is well known from universal algebra that $X^{+}$ and $X^{*}$ are unique up to isomorphism.