Definition:Word (Abstract Algebra)

Definition
Let $\struct {G, \circ}$ 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 $\map W S$:
 * $\map W S := \set {s_1 \circ s_2 \circ \cdots \circ s_n: n \in \N_{>0}: s_i \in S, 1 \le i \le n}$

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.

Also denoted as
Some sources use $\operatorname {gp} S$ for $\map W S$.

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