CounterexampleConstruction: AI-generatedVerification: 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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In $⟨ a, b ∣ a b, a b a ⟩$ , delete-only relator rewriting sends $a b a$ either to the empty word or to the stuck word $a$

Statement refuted

Use the following specific procedure: freely cancel adjacent inverse pairs or delete a contiguous occurrence of either displayed relator or its inverse, but never insert a relator and never lengthen the word.

The false claim is that the terminal string produced by this procedure is independent of the order of deletions. In $⟨ a, b ∣ a b, a b a ⟩$ , the word $a b a$ has one deletion path to the empty word and another to the stuck word $a$ .

Facts & Assumptions

Given: The presentation $G = ⟨ a, b ∣ a b, a b a ⟩$ , the word $a b a$ , and the delete-only procedure just stated.

[F1]

In a presented group, every displayed relator becomes the identity (Group presentation by generators and relations).

[F2]

Group multiplication is associative and has a two-sided identity (Group and abelian group).

Counterexample

technique · direct

1.1

The relations $a b = e$ and $a b a = e$ give $a = (a b) a = e$ and then $b = e$ , so every word represents the identity in $G$ .

F1F2

1.2

Deleting the occurrence of the relator $a b a$ from the whole word $a b a$ gives the empty word.

given

1.3

Deleting the prefix $a b$ gives the one-letter word $a$ , which has no inverse pair and contains none of $a b$ , $a b a$ , $b^{- 1} a^{- 1}$ , or $a^{- 1} b^{- 1} a^{- 1}$ ; hence it is stuck.

given

2.1

The two allowed first deletions end at different terminal strings, $ε$ and $a$ , even though step 1.1 shows that both represent the same group element; therefore the procedure is order-dependent.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Nothing in the library uses this result yet.

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.

In ⟨a,b∣ab, aba⟩⟨a,b∣ab, aba⟩, delete-only relator rewriting sends abaaba either to the empty word or to the stuck word aa