isIsomorphicTo
P29599
predicate
Indicates that two structures have a one-to-one, structure-preserving correspondence between their elements, making them equivalent in form even if not identical in content.
All labels observed (15)
| Label | Occurrences |
|---|---|
| isIsomorphicTo canonical | 27 |
| isomorphicTo | 18 |
| yieldsIsomorphism | 4 |
| givesIsomorphismBetween | 3 |
| hasCenterIsomorphicTo | 3 |
| centerIsIsomorphicTo | 2 |
| describesAsIsomorphic | 2 |
| expressesAsIsomorphism | 2 |
| groupIsIsomorphicTo | 2 |
| assertsIsomorphism | 1 |
| centerIsomorphicTo | 1 |
| hasComparisonIsomorphismWith | 1 |
| hasIsomorphism | 1 |
| isIsomorphicOverAlgebraicClosureTo | 1 |
| isomorphismType | 1 |
Description generation (PDg)
The one-sentence description above was generated by prompting gpt-5.1 with the predicate name and this instruction.
Instruction
Given a predicate that represents a relationship or action between entities, generate a one-sentence description explaining its meaning. # Instructions Focus on describing the relationship, not the entities themselves. # Response Format Begin the description with \' Indicates...\'
Input
Predicate: isIsomorphicTo
Generated description
Indicates that two structures have a one-to-one, structure-preserving correspondence between their elements, making them equivalent in form even if not identical in content.
Sample triples (69)
| Subject | Object |
|---|---|
| Lorentz group | SO^+(1,3) ⓘ |
| Klein quartic | modular curve X(7) over C ⓘ |
| AdS isometry group SO(2,d) | conformal group in d-dimensional Minkowski space ⓘ |
|
PSL(2,ℤ)
linked to:
modular group PSL(2,Z)
|
free product C₂ * C₃ via predicate surface "isomorphicTo" ⓘ |
| SU(3) | ℤ3 via predicate surface "hasCenterIsomorphicTo" ⓘ |
|
SO(3)
linked to:
rotation group SO(3)
|
group of orientation-preserving isometries of S² via predicate surface "isomorphicTo" ⓘ |
| SL(2,C) | Z/2Z via predicate surface "hasCenterIsomorphicTo" ⓘ |
| SL(2,C) |
Spin^+(3,1)
ⓘ
linked to:
rotation group SU(2)
|
| Harish-Chandra isomorphism | center of U(g) and S(h)^W via predicate surface "givesIsomorphismBetween" ⓘ |
| étale cohomology | singular cohomology over complex numbers via predicate surface "hasComparisonIsomorphismWith" ⓘ |
| PSL(2,7) |
GL(3,2)
via predicate surface "isomorphicTo"
ⓘ
linked to:
PSL(2,7)
|
| PSL(2,7) | L_2(7) via predicate surface "isomorphicTo" ⓘ |
| PSL(2,7) |
PSL_2(7)
via predicate surface "isomorphicTo"
ⓘ
linked to:
PSL(2,7)
|
| PSL(2,7) |
projective linear group of 2×2 matrices over F_7 with determinant 1 modulo scalars
via predicate surface "isomorphicTo"
ⓘ
linked to:
PSL(2,7)
|
| Fano plane | projective plane over GF(2) ⓘ |
|
su(2)
linked to:
rotation group SU(2)
|
so(3) ⓘ |
|
SU(2)
linked to:
rotation group SU(2)
|
Z/2Z via predicate surface "centerIsIsomorphicTo" NERFINISHED ⓘ |
|
SU(2)
linked to:
rotation group SU(2)
|
Spin(3) NERFINISHED ⓘ |
|
SU(2)
linked to:
rotation group SU(2)
|
unit quaternions ⓘ |
| ISO(n) | O(n) ⋉ R^n via predicate surface "isomorphicTo" NERFINISHED ⓘ |
|
SO(1)
linked to:
special orthogonal group SO(n)
|
{1} ⓘ |
|
SO(2)
linked to:
special orthogonal group SO(n)
|
U(1) NERFINISHED ⓘ |
|
SO(2)
linked to:
special orthogonal group SO(n)
|
circle group S¹ ⓘ |
|
SO(3)
linked to:
special orthogonal group SO(n)
|
projective special unitary group PSU(2) NERFINISHED ⓘ |
|
Dolbeault cohomology class
linked to:
Dolbeault cohomology classes
|
sheaf cohomology group H^q(X,Ω^p_X) on complex manifolds via predicate surface "isomorphicTo" ⓘ |
| de Rham cohomology | singular cohomology with real coefficients for smooth manifolds via predicate surface "isomorphicTo" ⓘ |
| de Rham cohomology | H^k_{dR}(M) ≅ H^k_{sing}(M;ℝ) via predicate surface "isomorphismType" ⓘ |
| U(1) | SO(2) NERFINISHED ⓘ |
| U(1) | R/Z ⓘ |
| U(1) | the circle group NERFINISHED ⓘ |
| SO(2,d-1) | SO(2,3) for d=4 ⓘ |
| SO(2,d-1) | SO(2,4) for d=5 ⓘ |
|
GL(n,ℂ)
linked to:
general linear group GL(n,C)
|
ℂˣ via predicate surface "centerIsomorphicTo" ⓘ |
|
SL(n,ℂ)
linked to:
special linear group SL(n,C)
|
μₙ (group of n-th roots of unity) via predicate surface "hasCenterIsomorphicTo" ⓘ |
| Verdier duality | RHom(F, D_X G) ≅ RHom(Rf_! F, G) via predicate surface "expressesAsIsomorphism" ⓘ |
| Verdier duality | H_c^i(X, F) ≅ H^{-i}(X, D_X F)^∨ under finiteness conditions via predicate surface "expressesAsIsomorphism" ⓘ |
| Lefschetz duality | H_i(M,\partial M) and H^{n-i}(M) via predicate surface "givesIsomorphismBetween" ⓘ |
| Lefschetz duality | H_i(M) and H^{n-i}(M,\partial M) via predicate surface "givesIsomorphismBetween" ⓘ |
| Birkhoff’s representation theorem for finite distributive lattices | finite distributive lattice via predicate surface "describesAsIsomorphic" ⓘ |
| Birkhoff’s representation theorem for finite distributive lattices | lattice of lower ideals of a finite poset via predicate surface "describesAsIsomorphic" ⓘ |
| Mordell curve | curve y^2 = x^3 + 1 (for k ≠ 0) via predicate surface "isIsomorphicOverAlgebraicClosureTo" ⓘ |
| GF(p) | Z/pZ ⓘ |
| GF(p) | prime field of characteristic p ⓘ |
| PSL(2,ℤ/Nℤ) | A₅ when N = 5 ⓘ |
| SL(2,ℤ) | free product C₄ *_{C₂} C₆ via predicate surface "isomorphicTo" ⓘ |
| PSL(2,ℝ) | orientation-preserving isometry group of hyperbolic plane via predicate surface "isomorphicTo" ⓘ |
| PSL(2,ℝ) | Isom⁺(ℍ²) via predicate surface "isomorphicTo" NERFINISHED ⓘ |
| Pontryagin duality | canonical evaluation map from a group to its double dual is an isomorphism via predicate surface "assertsIsomorphism" ⓘ |
| sl(2,C) | so(3,C) ⓘ |
| SL(2,R) | SU(1,1) as real Lie groups via predicate surface "isomorphicTo" ⓘ |