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.

Rigidity of two-letter words below braid length

Statement

Let $M$ be the positive monoid of a spherical Artin system with finite standard generating set $S$ and Coxeter matrix $(m_{s t})$ . Fix distinct $a, b \in S$ with braid parameter $m = m_{a b} < \infty$ . Every word supported in ${a, b}$ of length less than $m$ is its sole positive word representative in $M$ .

Statement references

Braid congruence preserves word length and support

Artin monoid and Artin group presentations, and the canonical monoid-to-group map

Facts & Assumptions

[F1]

Positive word equality is a sequence of support- and length-preserving braid substitutions. (Braid congruence preserves word length and support)

[F2]

The words and the standard Artin monoid presentation are defined in Artin monoid and Artin group presentations, and the canonical monoid-to-group map, Definition (1)–(3). For distinct $s, t \in S$ with $m_{s t} < \infty$ , the defining braid pair consists of the two alternating words of length $m_{s t} \geq 2$ , both with support ${s, t}$ ; an infinite parameter supplies no pair.

Proof

Given: The hypotheses and notation in the Statement.

Goal: Establish the asserted conclusion.

1.1F1F2given

Any substitution starting from such a word must use a relation supported in ${a, b}$ . By the defining braid pairs, the only possible relation is the one for the distinct pair $a, b$ , of length $m$ ; relations involving other generators cannot be applied because their left and right sides both contain those generators.

2.1F1step 1.1∎

The word is too short to contain a side of the length- $m$ relation. Thus no first substitution can change it, and the finite-sequence criterion implies that its equality class consists only of itself.

Depends on

Used by · 2

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.