Semiring Birkhoff factorization on the Connes–Kreimer Hopf algebra [UPSTREAM] #
The semiring form of [MCB25]'s renormalization (Def. 3.1.2, Prop. 3.1.9):
the linguistically operative case, where the target ℛ is a commutative semiring whose addition
is not invertible — tropical (ℝ ∪ {−∞}, max, +), Viterbi, and Boolean parsing semirings (§3.5,
"Birkhoff Factorization and (Semi)ring Parsing"; §3.5.2, "Minimal Yield as Birkhoff Factorization").
The Hopf algebra H = ConnesKreimer R (Nonplanar α) of nonplanar rooted forests is unchanged (base
R only a commutative semiring — the antipode-free factorization needs no negation, so this works
over R = ℕ, the base for a Boolean-semiring target), so the entire coproduct/cut infrastructure is
reused. Only the character target ℛ is a semiring, with a weight-+1 RotaBaxterSemiring
operator R. The
Bogolyubov recursion (Prop. 3.1.9, eq. (3.1.7)) reads
φ̃(x) = φ(x) ⊡ Σ φ₋(x′) ⊙ φ(x″), φ₋(x) = R(φ̃(x)), φ₊(x) = φ₋(x) ⊡ φ̃(x),
with ⊡, ⊙ the semiring addition/multiplication and R the positive projection (contrast the
ring case φ₋ = −R(φ̃), φ₊ = (1−R)(φ̃)). Because a semiring has no antipode, only the form
φ₊ = φ₋ ⋆ φ (Def. 3.1.6) is available — there is no φ = (φ₋ ∘ S) ⋆ φ₊ (Def. 3.1.5).
Main definitions #
birkhoffMinusTree φ R T/birkhoffMinus φ R: the Bogolyubov negative partφ₋ = R(φ̃)on a tree, and as an algebra homH →ₐ[R] ℛ.birkhoffPrepTree φ R T: the Bogolyubov preparationφ̃.birkhoffPlusTree φ R T: the renormalized partφ₊ = φ̃ + φ₋.
Main results #
birkhoffFactorization_ofTree:φ₊ = φ₋ ⋆ φon generators (Def. 3.1.6, Prop. 3.1.9 eq. (3.1.7)).
References #
[MCB25] (Def. 3.1.2, Def. 3.1.6, Prop. 3.1.9, Rem. 3.1.10)
The Bogolyubov negative part φ₋ on a single tree (semiring, weight +1;
[MCB25] Prop. 3.1.9):
φ₋(T) = R(Σ_{(cf,rem) ∈ cutSummandsN T} (Π_{Tᵢ ∈ cf} φ₋(Tᵢ)) · φ(ofTree rem)). The semiring
analogue of the ring birkhoffMinusTree, with the positive projection R in place of −R;
well-founded on T.depth.
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φ₋ extended multiplicatively to forests, as a MonoidHom. Mirrors the ring
birkhoffMinusMonoidHom.
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φ₋ as an algebra hom H →ₐ[R] ℛ, lifting birkhoffMinusMonoidHom. Mirrors the ring
birkhoffMinus.
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The Bogolyubov preparation φ̃ ([MCB25] Prop. 3.1.9):
φ̃(T) = Σ_{(cf,rem) ∈ cutSummandsN T} (Π_{Tᵢ ∈ cf} φ₋(Tᵢ)) · φ(ofTree rem), of which the
negative part is φ₋(T) = R(φ̃(T)) and the renormalized part is φ₊(T) = φ̃(T) + φ₋(T).
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The Bogolyubov preparation in non-attach-decorated form. Mirrors the ring
birkhoffPrepTree_unfold.
φ₋(T) = R(φ̃(T)): the negative part is the positive projection R of the preparation.
The renormalized part φ₊ on a single tree ([MCB25] Prop. 3.1.9):
φ₊(T) = φ̃(T) + φ₋(T) (the semiring φ₋ ⊡ φ̃) — the consistency-checked value.
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Semiring Birkhoff factorization on generators ([MCB25] Def. 3.1.6,
Prop. 3.1.9 eq. (3.1.7), φ₊ = φ₋ ⋆ φ): on each tree generator the convolution φ₋ ⋆ φ —
mul' ∘ (φ₋ ⊗ φ) ∘ comul — recovers the renormalized part φ₊. Needs φ unital (φ 1 = 1).
Same proof as the ring keystone (the identity is pure coproduct bookkeeping, sign-agnostic).