Definition:Word (Abstract Algebra)/Monoid

Definition
Let $\struct {M, \circ}$ be a monoid whose identity element is $e$.

Let $S \subseteq M$ be a subset of $M$.

The set of words in $S$ is denoted and defined:
 * $\map W S := \set {\ds \sum_{i \mathop = 1}^r n_i \cdot s_i : r \in \N, n_i \in \N, s_i \in S}$

where:
 * $n_i \cdot s_i$ denotes the power of $s_i$:
 * $n \cdot a = \begin {cases}

e & : n = 0 \\ \paren {\paren {n - 1} \cdot a} \circ a & : n > 0 \end {cases}$
 * $\ds \sum_{i \mathop = 1}^r n_i \cdot s_i := \paren {n_1 \cdot s_1} \circ \paren {n_2 \cdot s_2} \circ \cdots \circ \paren {n_r \cdot s_r}$

Also see

 * Definition:Generated Subgroup
 * Definition:Group Word on Set