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Transport of a Garside structure to the presented monoid

Statement

Let $j : M \to A^{+}$ be the positive-image isomorphism associated with an injective canonical map of an Artin monoid. Suppose $(A^{+}, ϕ^{+}, Δ^{+})$ is a one-object categorical Garside structure and $j (Δ) = Δ^{+}$ . Put $ϕ = j^{- 1} ϕ^{+} j$ . Then $(M, ϕ, Δ)$ is an atomic categorical Garside structure. Left and right gcds and lcms transfer through $j$ , and $x$ is simple in $M$ exactly when $j (x)$ is simple in $A^{+}$ . Here left and right gcds and lcms mean meets and joins in the respective divisibility orders.

Statement references

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

Divisibility and atomicity in a category

Categorical Garside structures and simple arrows

Positive degree and atoms of the presented Artin monoid

Identifying the presented monoid with its positive image

Facts & Assumptions

[F1]

A one-object atomic categorical Garside structure consists of an atomic cancellative monoid, an automorphism $ϕ$ with $x Δ = Δ ϕ (x)$ , simple atoms, and lattice orders for left and right divisibility; simple means $x y = Δ$ for some $y$ . (Categorical Garside structures and simple arrows, Divisibility and atomicity in a category)

[F2]

The isomorphism $j$ and its inverse preserve products, both divisibility relations and atoms. (Identifying the presented monoid with its positive image)

[F3]

The presented Artin monoid has factorizations into atoms and a degree bounding all nonidentity factorizations. (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.

1.1F1F2F3given

Conjugating $ϕ^{+}$ by the isomorphism yields a monoid automorphism $ϕ$ . Applying the injective $j$ to the desired naturality equation gives $j (x) Δ^{+} = Δ^{+} ϕ^{+} (j (x))$ , which holds by the assumed structure. Cancellation in $M$ follows by applying $j$ and using cancellation in $A^{+}$ . Atomicity holds by [F3], including bounded lengths of all nonidentity factorizations.

2.1F1F2step 1.1∎

By [F2] each divisibility order is order-isomorphic to its counterpart in $A^{+}$ . The inverse image of the meet or join satisfies exactly its defining universal property, giving both gcds and lcms. An equality $j (x) y^{+} = Δ^{+}$ pulls back uniquely to $x y = Δ$ ; therefore simplicity transfers. Atoms transfer by [F2], so each is simple since atoms in $A^{+}$ are simple. Together these verify every condition in [F1].

Depends on

Used by · 1

Independent spherical Artin Garside input

Dependency tree · 2 levels

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