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.

Identifying the presented monoid with its positive image

Statement

For a finite-rank Coxeter system $(W, S)$ , let $j : M \to A$ be the canonical homomorphism from the presented positive Artin monoid to its Artin group. Let $A^{+}$ denote the submonoid of $A$ generated by the positive generators. If $j$ is injective, it is an isomorphism $M \to A^{+}$ . It carries $w$ to $σ_{w}$ and, in finite type (that is, when $W$ is finite), $w_{0}$ to $σ_{w_{0}}$ , where $w_{0}$ is the unique element of maximal Coxeter length. A factorization in $A^{+}$ holds exactly when its unique preimages factor in $M$ . In particular both divisibility relations, atoms and the additive degree are transported by this isomorphism.

Statement references

Positive degree and atoms of the presented Artin monoid

Reduced Coxeter lifts in the presented positive monoid

The longest element as the opposition of the chamber, and longest elements of finite parabolics

Facts & Assumptions

[F1]

The map $j$ sends each positive monoid generator to the corresponding Artin group generator and satisfies $j (w) = σ_{w}$ . (Reduced Coxeter lifts in the presented positive monoid)

[F2]

The degree $d$ and the atoms of $M$ are as described in the positive-degree lemma. (Positive degree and atoms of the presented Artin monoid)

[F3]

If $W$ is finite, it has a unique element $w_{0}$ of maximal Coxeter length. (The longest element as the opposition of the chamber, and longest elements of finite parabolics, Statement (1)(iii))

Proof

Given: The objects and hypotheses in the statement.

Goal: Establish all assertions of the statement.

1.1F1F3given

Every element of $j (M)$ is a product of positive generators, and every such product is the image of its word in $M$ . Hence $j (M) = A^{+}$ . Under the injectivity hypothesis, $j$ is a bijective homomorphism onto $A^{+}$ ; its inverse respects products because the preimage of $j (x) j (y)$ is $x y$ . The lift assertion follows from [F1]. In finite type, [F3] supplies $w_{0}$ , so applying [F1] to $w_{0}$ also proves the longest-lift assertion.

2.1F2step 1.1∎

For $x, y, z \in M$ , equality $j (x) j (y) = j (z)$ is equivalent by injectivity to $x y = z$ . Left and right divisibility are existence assertions of these equalities, and atoms are nonidentity elements without such a factorization into two nonidentity factors, so all these properties transfer. Finally $d^{+} (j (x)) = d (x)$ is well defined by bijectivity and is additive by [F2].

Depends on

Used by · 3

Dependency tree · 2 levels

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