PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-11

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Free equivalence is an equivalence relation and concatenation respects it

Statement

For words on $X ⊔ X^{- 1}$ , free equivalence is an equivalence relation in the sense of Equivalence relation, equivalence class, and the quotient set $A / \sim$ . It is also a congruence for concatenation: if $w \sim w^{'}$ and $v \sim v^{'}$ , then $w v \sim w^{'} v^{'}$ .

Facts & Assumptions

Given: A set $X$ and finite words $u, v, w, w^{'}, v^{'}$ on $X ⊔ X^{- 1}$ .

[F1]

Words are freely equivalent if one can be transformed into the other by finitely many elementary cancellations and their reverse insertions (Words in an alphabet with formal inverses, elementary cancellation, and reduced words).

Proof

technique · direct

1.1

The empty sequence of elementary moves carries every word $w$ to itself, so $w \sim w$ .

1.2

If a finite sequence of elementary moves carries $w$ to $w^{'}$ , reversing its order and interchanging every cancellation with the corresponding insertion gives a finite sequence from $w^{'}$ to $w$ ; hence $w \sim w^{'}$ implies $w^{'} \sim w$ .

1.3

If $w \sim w^{'}$ and $w^{'} \sim w^{''}$ , concatenating the two finite move sequences gives a finite move sequence from $w$ to $w^{''}$ ; hence free equivalence is transitive.

1.4

If one elementary move changes $w$ to $w^{'}$ , then the same adjacent pair can be deleted or inserted inside $u w v$ , so the move changes $u w v$ to $u w^{'} v$ ; applying this to every move in a finite sequence gives $w \sim w^{'} \Rightarrow u w v \sim u w^{'} v$ .

2.1

If $w \sim w^{'}$ and $v \sim v^{'}$ , step 1.4 gives $w v \sim w^{'} v$ and $w^{'} v \sim w^{'} v^{'}$ ; transitivity gives $w v \sim w^{'} v^{'}$ . Thus steps 1.1 through 1.3 prove that $\sim$ is an equivalence relation, and this step proves the congruence claim.

step 1.1step 1.2step 1.3step 1.4∎

Depends on

Used by

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, §1.2 (standard reference, not scraped)