definition 4.4 Internal binary operation; magma

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:115 · p. 58

ground object -- no derivation owed

Rests on

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 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶ ↺
depends_on definition 4.2 Binary operation ¶ ↺

Supports

depends_on definition 4.34 Algebra ¶
depends_on definition 4.8 Commutativity; abelian structure ¶
- 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 5.101 External direct sum ¶
      - depends_on definition 5.18 Inner product ¶
      - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 5.19 Norm ¶
      - 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 $\Z_2\times\Z_4$ ¶ ↺
depends_on definition 4.18 Semigroup ¶
- depends_on definition 4.20 Monoid ¶
  - depends_on definition 4.32 Field ¶ ↺
  - depends_on definition 4.21 Group ¶
    - depends_on definition 4.37 Cayley table ¶
      - depends_on example 4.69 The Cayley table of $\Z_2\times\Z_4$ ¶ ↺
    - depends_on definition 4.65 Direct product of groups ¶
      - depends_on proposition 4.68 Direct product of abelian groups ¶ ↺
      - depends_on proposition 4.70 Characterization of the direct product ¶
      - depends_on proposition 4.66 prop:alg-direct-product-group ¶
      - depends_on proposition 4.77 When a semidirect product is direct ¶
    - depends_on definition 4.35 The group axioms, restated one at a time ¶
    - depends_on definition 4.43 Group homomorphism ¶
      - depends_on definition 4.44 Group isomorphism ¶
      - depends_on definition 4.73 External semidirect product ¶
      - depends_on definition 5.141 Representation of a group ¶
      - depends_on proposition 4.42 When the homomorphisms form a group ¶
      - depends_on proposition 4.47 Image of the inverse element ¶
      - depends_on proposition 4.46 Image of the neutral element ¶
      - depends_on proposition 4.17 Trivial kernel of a homomorphism ¶
    - depends_on definition 4.31 Module ¶ ↺
    - depends_on definition 4.49 Permutation group ¶
      - depends_on proposition 4.50 Cardinal of the permutation group ¶
      - depends_on proposition 4.53 The subgroups of $S_3$ ¶
      - depends_on theorem 4.56 Cayley's theorem ¶
    - depends_on definition 4.30 Ring ¶ ↺
    - depends_on definition 4.36 Subgroup ¶
      - depends_on definition 4.59 Conjugate subgroups ¶
      - depends_on definition 4.60 Normal subgroup ¶
      - depends_on definition 4.67 Product of subgroups ¶
      - depends_on proposition 4.72 The automorphism group ¶
      - depends_on proposition 4.61 prop:alg-coset-equiv ¶
      - depends_on proposition 4.42 When the homomorphisms form a group ¶ ↺
      - depends_on proposition 4.38 prop:alg-nZ-subgroup ¶
      - depends_on proposition 4.53 The subgroups of $S_3$ ¶ ↺
      - depends_on theorem 4.56 Cayley's theorem ¶ ↺
    - depends_on definition A.561 Coadjoint action and equivariance ¶
      - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
      - depends_on theorem A.563 Marsden–Weinstein reduction ¶
    - depends_on definition 5.141 Representation of a group ¶ ↺
    - depends_on example 4.22 A two-element group ¶
    - depends_on proposition 4.40 prop:alg-congruence-equiv ¶
      - depends_on proposition 4.41 prop:alg-zn-group ¶ ↺
    - … 11 more
- depends_on definition 4.30 Ring ¶ ↺
- depends_on example 4.19 A two-element semigroup ¶
- depends_on proposition 15.62 The complex line is not a semigroup algebra ¶
  - depends_on remark 15.63 The inclusion is strict, and the strictness is physical ¶
depends_on definition 4.12 Substructure ¶
depends_on definition 5.121 Algebra ¶ ↺

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`	→	Algebraic structure	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:124
`depends_on`	→	Binary operation	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:124
`depends_on`	←	Algebra	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:882
`depends_on`	←	Commutativity; abelian structure	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:171
`depends_on`	←	Semigroup	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:495
`depends_on`	←	Substructure	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:253
`depends_on`	←	Algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5258