Definition:Word (Abstract Algebra)

Definition
Let $$S \subseteq G$$ where $$\left({G, \circ}\right)$$ is an algebraic structure.

A word in $$S$$ is the product of a finite number of elements of $$S$$.

The set of words in $$S$$ is denoted $$W \left({S}\right)$$:
 * $$W \left({S}\right) \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{s_1 \circ s_2 \circ \cdots \circ s_n: n \in \N^*: s_i \in S, 1 \le i \le n}\right\}$$

Note that there is nothing in this definition preventing any of the elements of $$S$$ being repeated, neither is anything said about the order of these elements.

Some sources use $$\operatorname {gp} \, S$$ for $$W \left({S}\right)$$.

Context
It is usual for the algebraic structure in question to be a group or sometimes semigroup.

If the operation $$\circ$$ is not associative then this definition still holds.