Triple

T11219353
Position Surface form Disambiguated ID Type / Status
Subject Milnor K-theory E265518 entity
Predicate usesConcept P531 FINISHED
Object Steinberg relations
Steinberg relations are algebraic identities in Milnor K-theory that impose the condition that symbols of pairs of field elements summing to one vanish, playing a central role in defining the structure of these K-groups.
E911353 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
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).
Input
Phrase: Steinberg relations | Statement: [Milnor K-theory, usesConcept, Steinberg relations]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Steinberg relations
Context triple: [Milnor K-theory, usesConcept, Steinberg relations]
  • A. Dolan–Grady relations
    The Dolan–Grady relations are algebraic commutation relations between two operators that generate the Onsager algebra and play a key role in the study of exactly solvable models in statistical mechanics.
  • B. The Poincaré-Birkhoff-Witt theorem in ring theory
    "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.
  • C. Schur–Weyl duality
    Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
  • D. 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.
  • E. Zassenhaus conjecture
    The Zassenhaus conjecture is a prominent open problem in group theory concerning the structure of units in integral group rings and their relation to the underlying finite group.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Steinberg relations
Triple: [Milnor K-theory, usesConcept, Steinberg relations]
Generated description
Steinberg relations are algebraic identities in Milnor K-theory that impose the condition that symbols of pairs of field elements summing to one vanish, playing a central role in defining the structure of these K-groups.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Steinberg relations
Target entity description: Steinberg relations are algebraic identities in Milnor K-theory that impose the condition that symbols of pairs of field elements summing to one vanish, playing a central role in defining the structure of these K-groups.
  • A. Dolan–Grady relations
    The Dolan–Grady relations are algebraic commutation relations between two operators that generate the Onsager algebra and play a key role in the study of exactly solvable models in statistical mechanics.
  • B. The Poincaré-Birkhoff-Witt theorem in ring theory
    "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.
  • C. Schur–Weyl duality
    Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
  • D. 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.
  • E. Zassenhaus conjecture
    The Zassenhaus conjecture is a prominent open problem in group theory concerning the structure of units in integral group rings and their relation to the underlying finite group.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69d6aac59460819089b9848b27f57848 completed April 8, 2026, 7:21 p.m.
NER Named-entity recognition batch_69d7e8eb84c48190b4f3bede254afde2 completed April 9, 2026, 5:59 p.m.
NED1 Entity disambiguation (via context triple) batch_69e4976f38788190855aed6338d819b7 completed April 19, 2026, 8:50 a.m.
NEDg Description generation batch_69e49d37989881909c7e75ddfff06726 completed April 19, 2026, 9:15 a.m.
NED2 Entity disambiguation (via description) batch_69e49f41a1f8819087cc15527dc7ff63 completed April 19, 2026, 9:24 a.m.
Created at: April 8, 2026, 9:30 p.m.