Triple
T7338616
| Position | Surface form | Disambiguated ID | Type / Status |
|---|---|---|---|
| Subject | PSL(2,ℤ) |
E169191
|
entity |
| Predicate | hasQuotient |
P77178
|
FINISHED |
| Object |
PSL(2,ℤ/Nℤ)
PSL(2,ℤ/Nℤ) is the projective special linear group of 2×2 matrices with entries in the ring of integers modulo N, modulo scalar matrices, forming a fundamental example of a finite (or, for composite N, generally non-simple) group in algebra and number theory.
|
E656691
|
NE FINISHED |
How this triple was built (5 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: PSL(2,ℤ/Nℤ) | Statement: [PSL(2,ℤ), hasQuotient, PSL(2,ℤ/Nℤ)]
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: PSL(2,ℤ/Nℤ) Context triple: [PSL(2,ℤ), hasQuotient, PSL(2,ℤ/Nℤ)]
-
A.
modular group PSL(2,Z)
The modular group PSL(2,ℤ) is a fundamental discrete group of 2×2 integer matrices modulo sign, acting by fractional linear transformations on the upper half-plane and playing a central role in number theory, geometry, and the theory of modular forms.
-
B.
PSL(2,7)
PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
-
C.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
D.
Fuchsian group
A Fuchsian group is a discrete group of isometries of the hyperbolic plane, fundamental in the study of Riemann surfaces, modular forms, and hyperbolic geometry.
-
E.
special linear group SL(n,R)
The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
- 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: PSL(2,ℤ/Nℤ) Triple: [PSL(2,ℤ), hasQuotient, PSL(2,ℤ/Nℤ)]
Generated description
PSL(2,ℤ/Nℤ) is the projective special linear group of 2×2 matrices with entries in the ring of integers modulo N, modulo scalar matrices, forming a fundamental example of a finite (or, for composite N, generally non-simple) group in algebra and number theory.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: PSL(2,ℤ/Nℤ) Target entity description: PSL(2,ℤ/Nℤ) is the projective special linear group of 2×2 matrices with entries in the ring of integers modulo N, modulo scalar matrices, forming a fundamental example of a finite (or, for composite N, generally non-simple) group in algebra and number theory.
-
A.
modular group PSL(2,Z)
The modular group PSL(2,ℤ) is a fundamental discrete group of 2×2 integer matrices modulo sign, acting by fractional linear transformations on the upper half-plane and playing a central role in number theory, geometry, and the theory of modular forms.
-
B.
PSL(2,7)
PSL(2,7) is a finite simple group of order 168, notable as the full automorphism group of the Klein quartic and as a key example in the theory of projective linear groups over finite fields.
-
C.
SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
-
D.
Fuchsian group
A Fuchsian group is a discrete group of isometries of the hyperbolic plane, fundamental in the study of Riemann surfaces, modular forms, and hyperbolic geometry.
-
E.
special linear group SL(n,R)
The special linear group SL(n,ℝ) is the Lie group of all n×n real matrices with determinant 1, fundamental in linear algebra and differential geometry as the group of volume-preserving linear transformations.
- F. None of above. chosen
PD
Predicate disambiguation
gpt-5-mini-2025-08-07
Target predicate: hasQuotient Context triple: [PSL(2,ℤ), hasQuotient, PSL(2,ℤ/Nℤ)]
-
A.
isQuotientOf
Indicates that one quantity is the result of dividing another quantity by a specified divisor.
-
B.
hasDenominatorFactor
Indicates that one value appears as a factor in the denominator of another value or expression.
-
C.
hasFraction
Indicates that one entity represents a fractional part or proportion of another entity.
-
D.
dividedBy
Indicates that one quantity is separated into a specified number of equal parts or groups by another quantity, representing a division relationship between them.
-
E.
isDividedBy
Indicates that one quantity or entity serves as the divisor that evenly or proportionally separates another quantity or entity into parts.
- F. None of above. chosen
Provenance (7 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_69c68a57710481909f0c1f3c6ebdb6f2 |
completed | March 27, 2026, 1:47 p.m. |
| NER | Named-entity recognition | batch_69c6f347f25081908e6086d4073295f5 |
completed | March 27, 2026, 9:14 p.m. |
| NED1 | Entity disambiguation (via context triple) | batch_69c7ef266fd0819096cf3ece3fff6b90 |
completed | March 28, 2026, 3:09 p.m. |
| NEDg | Description generation | batch_69c7efa4f5148190842f30988cbea94c |
completed | March 28, 2026, 3:11 p.m. |
| NED2 | Entity disambiguation (via description) | batch_69c7f0092bac819080ded1863f99290a |
completed | March 28, 2026, 3:13 p.m. |
| PD | Predicate disambiguation | batch_69c6f028fd748190b2ea5c3081958a42 |
completed | March 27, 2026, 9:01 p.m. |
| PDg | Predicate description generation | batch_69c6f3463d0481908aed9ed43a8ac6a8 |
completed | March 27, 2026, 9:14 p.m. |
Created at: March 27, 2026, 3:04 p.m.