Definition:Word (Abstract Algebra)

Definition
Let $\left({G, \circ}\right)$ be a magma.

Let $S \subseteq G$ be a subset.

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

The set of words in $S$ is denoted $W \left({S}\right)$:
 * $W \left({S}\right) := \left\{{s_1 \circ s_2 \circ \cdots \circ s_n: n \in \N_{>0}: s_i \in S, 1 \le i \le n}\right\}$

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.

Also see

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