LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The two-generator braid element is the lcm

Statement

In a spherical positive Artin monoid, for distinct standard generators $a, b$ and $m = m_{a b}$ , their least common multiple for left divisibility is the element represented by the two alternating words of length $m$ . These are exactly its two positive word representatives.

Statement references

Independent spherical Artin Garside input

Rigidity of two-letter words below braid length

Exactly two words for a two-generator braid element

Facts & Assumptions

[F1]

The positive monoid is cancellative and has a left-divisibility lcm for every pair. (Independent spherical Artin Garside input)

[F2]

Words on $a, b$ shorter than $m$ have only one representative. (Rigidity of two-letter words below braid length)

[F3]

The length- $m$ braid element $d$ has exactly its two alternating representatives. (Exactly two words for a two-generator braid element)

Proof

Given: The hypotheses and notation in the Statement.

Goal: Establish the asserted conclusion.

1.1F1F3

Both $a$ and $b$ left-divide $d$ , so their lcm $c$ left-divides $d$ . Choose positive words $p, q$ for $c$ and its complement. Then $p q$ is one of the two alternating length- $m$ words by F3. In particular $p$ is a prefix of one of these words.

2.1F1F2F3step 1.1∎

If $∣ p ∣ < m$ , F2 makes $p$ the unique word representing $c$ . But $a$ and $b$ both left-divide $c$ , giving word representatives of $c$ starting with these distinct letters, a contradiction. Thus $∣ p ∣ = m$ and $q$ is empty, so $c = d$ . The representative count is F3.

Depends on

Used by · 2

Dependency tree · 2 levels

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