Alphabeta Math
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.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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In ⟨a,b∣ab⟩, the trivial word ba is stuck under free cancellation and delete-only relator rewriting

Statement refuted

Consider the following specific syntactic procedure for a displayed finite presentation: at each step, freely cancel an adjacent inverse pair or delete a contiguous occurrence of one displayed relator or its inverse. Never insert a relator and never lengthen the word.

The false claim is that this delete-only procedure reduces every word that represents the identity to the empty word. In ⟨a,b∣ab⟩, the word ba represents the identity but admits no step at all.

Facts & Assumptions

Given: The presentation G=⟨a,b∣ab⟩ 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]

In a group, an equation yx=e determines y=x−1 (Group and abelian group).

Counterexample

technique · direct
1.1F1F2

The relation ab=e gives b=a−1 by [F2], and therefore ba=a−1a=e in G.

1.2given

The word ba has no adjacent inverse pair and contains neither the relator ab nor its inverse b−1a−1 as a contiguous subword.

2.1step 1.1step 1.2∎

Thus ba represents the identity by step 1.1 but is stuck and nonempty under the stated procedure by step 1.2, refuting the claim.

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.