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.

Exactly two words for a two-generator braid element

Statement

Let $S$ be any set and $(m_{s t})_{s, t \in S}$ a symmetric matrix with $m_{s s} = 1$ and $m_{s t} \in {2, 3, \dots} \cup {\infty}$ for $s \neq t$ . Let $M$ be its presented positive Artin monoid: finite words on $S$ modulo the congruence generated by the standard alternating braid pairs, as constructed in the braid-congruence lemma. Write $[w]$ for the element represented by a word $w$ . Fix distinct $a, b \in S$ with $m = m_{a b} < \infty$ . Let $u$ and $v$ be the alternating length- $m$ words starting with $a$ and $b$ , respectively. The element $d = [u] = [v]$ has exactly the two positive word representatives $u, v$ in $M$ . No spherical assumption is needed for this presented monoid.

Statement references

Braid congruence preserves word length and support

Facts & Assumptions

[F1]

For the arbitrary-set presented monoid, positive word equality is equivalent to a finite sequence of braid substitutions, preserving length and support. Its construction defines the alternating braid pairs: each uses a distinct pair $s, t$ , has length $m_{s t} \geq 2$ and has support ${s, t}$ on both sides; an infinite parameter supplies no pair. (Braid congruence preserves word length and support)

Proof

Given: The hypotheses and notation in the Statement.

Goal: Determine all positive words representing $d$ .

1.1F1given

The defining $a, b$ braid relation gives $[u] = [v]$ , and the two words are distinct because their initial letters differ. Since $m \geq 2$ , both have support ${a, b}$ . Every word in a substitution sequence starting at $u$ has length $m$ and support ${a, b}$ .

2.1F1step 1.1∎

Any applicable braid relation uses a distinct pair $s, t$ whose two letters must both occur in the current word. Thus ${s, t} = {a, b}$ , and the substituted side has length $m$ . Since the current word also has length $m$ , both contexts must be empty. The substitution therefore replaces the entire word $u$ by $v$ or $v$ by $u$ . By induction on the number of substitutions every word reachable from $u$ is $u$ or $v$ . F1 identifies reachability with equality in $M$ , and step 1.1 gives both representatives, proving the assertion.

Depends on

Braid congruence preserves word length and support

Used by · 2

Dependency tree · 2 levels

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