definition 4.8 Commutativity; abelian structure

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:164 · p. 58

ground object -- no derivation owed

Rests on

depends_on definition 4.4 Internal binary operation; magma ¶
- depends_on definition 4.3 Algebraic structure ¶
  - depends_on definition 4.2 Binary operation ¶
    - depends_on definition 4.1 Cartesian product ¶
      - depends_on definition 3.28 Set ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶ ↺
- depends_on definition 4.2 Binary operation ¶ ↺

Supports

depends_on definition 4.30 Ring ¶
- depends_on definition 4.32 Field ¶
  - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.34 Algebra ¶
    - depends_on definition 5.121 Algebra ¶
      - depends_on definition 5.128 Derivation ¶
      - depends_on definition 5.124 Ideal ¶
      - depends_on definition 5.127 Lie algebra ¶
      - depends_on definition 5.122 Linear hull ¶
      - depends_on definition 5.123 Subalgebra ¶
      - depends_on definition 5.162 Representation of an algebra ¶
      - depends_on proposition 5.125 prop:lin-matrix-algebra ¶
    - depends_on definition 5.101 External direct sum ¶
      - depends_on definition 12.84 External direct sum ¶
      - depends_on definition 5.103 Sum of subspaces; internal direct sum ¶
      - depends_on proposition 5.102 Dimensions add ¶
    - depends_on definition 5.18 Inner product ¶
      - depends_on corollary 5.29 Orthogonal decomposition ¶
      - depends_on definition 7.96 Plane ¶
      - depends_on definition 12.2 Hilbert space ¶
      - depends_on definition 12.94 Tensor product of Hilbert spaces ¶
      - depends_on definition 5.41 Adjoint ¶
      - depends_on definition 5.32 Levi–Civita symbol; cross product ¶
      - depends_on definition 5.24 Orthogonal vectors ¶
      - depends_on definition 5.144 Unitary representation ¶
      - depends_on example 12.9 The sequence space $\ell^{2}$ ¶
      - depends_on example 12.11 The function space $L^{2}$ ¶
      - depends_on proposition 12.6 Parallelogram law and polarization ¶
      - depends_on proposition 5.63 prop:lin-dual-inner-product ¶
      - … 5 more
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 5.101 External direct sum ¶ ↺
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.7 Vector subspace ¶
      - depends_on example 5.8 The coordinate spaces $\R^{n}$ and $\C^{n}$ ¶
      - depends_on proposition 5.107 The union of two subspaces ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶
      - depends_on definition 12.69 Operator with a domain ¶
      - depends_on definition 5.41 Adjoint ¶ ↺
      - depends_on definition 5.110 Bilinear map ¶
      - depends_on definition 5.128 Derivation ¶ ↺
      - depends_on definition 5.56 Endomorphism ¶
      - depends_on definition 5.46 Functional ¶
      - depends_on definition 5.47 Inverse of a linear transformation ¶
      - depends_on definition 5.53 Isomorphism ¶
      - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.66 Pushforward ¶
      - … 6 more
    - depends_on definition 5.19 Norm ¶
      - depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
      - depends_on definition 5.22 Metric associated with a norm ¶
      - depends_on definition 5.25 Unit vector ¶
      - depends_on definition 9.21 Matrix exponential ¶
      - depends_on proposition 12.8 Absolutely convergent series test ¶
    - depends_on definition 5.3 Scalar ¶
    - depends_on definition 5.115 Sesquilinear form ¶
    - depends_on definition 5.4 Vector ¶
    - depends_on example 5.8 The coordinate spaces $\R^{n}$ and $\C^{n}$ ¶ ↺
    - depends_on example 5.10 Functions on a set ¶
    - … 1 more
  - depends_on lemma A.44 $\R$ is an ordered field ¶
  - depends_on theorem A.35 Completeness of $\R$ ¶
- depends_on definition 4.31 Module ¶
  - depends_on definition 4.33 Vector space ¶ ↺
depends_on definition 4.33 Vector space ¶ ↺
depends_on proposition 4.68 Direct product of abelian groups ¶
- depends_on example 4.69 The Cayley table of $\Z_2\times\Z_4$ ¶
depends_on proposition 4.52 $S_3$ is a smallest non-abelian group ¶
depends_on proposition 4.41 prop:alg-zn-group ¶
- depends_on example 4.69 The Cayley table of $\Z_2\times\Z_4$ ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Internal binary operation; magma	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:171
`depends_on`	←	Ring	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:773
`depends_on`	←	Vector space	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:848
`depends_on`	←	Direct product of abelian groups	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2446
`depends_on`	←	$S_3$ is a smallest non-abelian group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1819
`depends_on`	←	prop:alg-zn-group	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:1359