Triple

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Phrase: Poincaré–Birkhoff–Witt theorem | Statement: [The Poincaré-Birkhoff-Witt theorem in ring theory, hasTheoremAsTopic, Poincaré–Birkhoff–Witt theorem]

Target entity: Poincaré–Birkhoff–Witt theorem
Context triple: [The Poincaré-Birkhoff-Witt theorem in ring theory, hasTheoremAsTopic, Poincaré–Birkhoff–Witt theorem]

• A. The Poincaré-Birkhoff-Witt theorem in ring theory chosen
  "The Poincaré-Birkhoff-Witt theorem in ring theory" is a mathematical work, attributed here to N. G. de Bruijn, that studies and applies the Poincaré–Birkhoff–Witt theorem in the context of associative and Lie-theoretic ring structures.
• B. Eilenberg–Zilber theorem
  The Eilenberg–Zilber theorem is a fundamental result in algebraic topology that establishes a chain homotopy equivalence between the singular chain complex of a product space and the tensor product of the singular chain complexes of the factors.
• C. Weyl algebra
  The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
• D. Hilbert’s syzygy theorem
  Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
• E. Rota–Baxter algebra
  A Rota–Baxter algebra is an associative algebra equipped with a linear operator satisfying a specific integration-like identity that generalizes the properties of integral and summation operators in algebraic form.
• F. None of above.
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target predicate: hasTheoremAsTopic
Context triple: [The Poincaré-Birkhoff-Witt theorem in ring theory, hasTheoremAsTopic, Poincaré–Birkhoff–Witt theorem]

• A. hasTheorem chosen
  Indicates that one entity (typically a mathematical theory, field, or work) includes, establishes, or is associated with a particular theorem.
• B. hasTheory
  Indicates that an entity possesses, is associated with, or is characterized by a particular theory.
• C. hasTheoremNamedAfter
  Indicates that a theorem is named in honor of or after a particular person or entity.
• D. relatedTheorem
  Indicates that one theorem is connected to another through a logical, thematic, or derivational relationship.
• E. hasCorollary
  Indicates that one statement, result, or proposition follows as a corollary from another, typically more general, statement.
• F. None of above.