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.

Reduced Coxeter lifts in the presented positive monoid

Statement

Let $(W, S)$ be a finite-rank Coxeter system with length $ℓ$ , let $M$ be its presented positive Artin monoid, and let $A$ be its Artin group. Write $q : M \to W$ and $j : M \to A$ for the generator maps. For a reduced expression $w = s_{1} \dots s_{r}$ , set $w = a_{s_{1}} \dots a_{s_{r}}$ in $M$ . This is independent of the reduced expression, $q (w) = w$ , and $d (w) = ℓ (w)$ . Thus $w \mapsto w$ is injective as a map of sets. Whenever $ℓ (u v) = ℓ (u) + ℓ (v)$ , one has $u v = u v$ . Moreover $j (w) = σ_{w}$ , the reduced-word lift in $A$ . No injectivity of $j$ is assumed here.

Statement references

Positive degree and atoms of the presented Artin monoid

Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups

Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups

Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares

Reduced-word lifts and the prefix-order convention

Facts & Assumptions

[F1]

Reduced words have length $ℓ (w)$ , and any two reduced words for the same Coxeter element are connected by braid moves. (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups)

[F2]

The generator maps $q$ and $j$ respect the defining braid relations. (Universal properties of the Artin monoid and group, the projection onto the Coxeter group, and the quotient by the squares)

[F3]

The degree $d$ is word length on the presented monoid. (Positive degree and atoms of the presented Artin monoid)

[F4]

The group lift $σ_{w}$ is the product of the corresponding generators along any reduced word for $w$ . (Reduced-word lifts and the prefix-order convention)

[F5]

The monoid $M$ is the quotient of the free monoid on the generators $a_{s}$ by the congruence generated by the braid relations. In particular, the two alternating words of length $m (s, t) < \infty$ are equal in $M$ , also inside any word context. (Positive degree and atoms of the presented Artin monoid)

Proof

Given: The objects and hypotheses in the statement.

Goal: Establish all assertions of the statement.

1.1F1F2F3F5

Any two reduced words for $w$ are connected by the moves in [F1], and each move is an equality in the presented monoid by [F5]. Therefore $w$ is well defined. Its image under $q$ is $w$ , and its degree is $ℓ (w)$ by [F3]. Applying $q$ shows that equal lifts have equal Coxeter elements, establishing set injectivity.

2.1F1F2F4step 1.1∎

Concatenate reduced words for $u$ and $v$ . If their lengths add to $ℓ (u v)$ , that concatenation is reduced and gives $u v = u v$ by step 1.1. Applying $j$ to a reduced word yields exactly the product defining $σ_{w}$ in [F4]. These arguments do not cancel elements in $M$ or assume that $j$ is injective.

Depends on

Used by · 4

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.