CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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In $\langle a,b\mid ab,\ aba\rangle$ , delete-only relator rewriting sends $aba$ 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 $\langle a,b\mid ab,aba\rangle$ , the word $aba$ has one deletion path to the empty word and another to the stuck word $a$ .

Facts & Assumptions

Given: The presentation $G=\langle a,b\mid ab,aba\rangle$ , the word $aba$ , 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 $ab=e$ and $aba=e$ give $a=(ab)a=e$ and then $b=e$ , so every word represents the identity in $G$ .

F1F2

1.2

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

given

1.3

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

given

2.1

The two allowed first deletions end at different terminal strings, $\varepsilon$ 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

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 19 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

In ⟨a,b∣ab, aba⟩\langle a,b\mid ab,\ aba\rangle⟨a,b∣ab, aba⟩, delete-only relator rewriting sends abaabaaba either to the empty word or to the stuck word aaa