Definition:Word (Abstract Algebra)
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This page is about the product of a finite number of elements of a given subset of an algebraic structure. For other uses, see Definition:Word.
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 $W \left({S}\right)$:
- $W \left({S}\right) := \left\{{s_1 \circ s_2 \circ \cdots \circ s_n: n \in \N^*: s_i \in S, 1 \le i \le n}\right\}$
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.
Sources
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 5.3$
