Group theory and generalizations Definition

Reduced word

Definition

Let $S$ be a set, and let $S^{-1} = \{s^{-1} : s \in S\}$ be a set of new symbols, one for each $s \in S$ . A word on $S$ is a finite string $t_1 t_2 \cdots t_n$ with $n \geq 0$ and each $t_i$ in $S \cup S^{-1}$ . A word is reduced when it contains no adjacent pair $s s^{-1}$ or $s^{-1} s$ with $s \in S$ .

Remarks

The empty word, with $n = 0$ , is reduced. Remove adjacent pairs $s s^{-1}$ and $s^{-1} s$ , one at a time, until no such pair is left. The result is a reduced word. For example, for $S = \{a, b, c\}$ , the word $a b^3 c^{-1} c a^{-1} c$ reduces to $a b^3 a^{-1} c$ . A reduced word is cyclically reduced when its first and last letters are not inverse to each other.

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Tag	`0FPXK5CHQ3`
URI	`m2://0FPXK5CHQ3`
Aliases	`freely-reduced-word`
Kind	definition
Writing	W0 stub
MSC	20E05
Defines	Reduced word

Reduced word

Definition

Remarks

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