definition 4.1 Cartesian product

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:47 · p. 57

ground object -- no derivation owed

Rests on

depends_on definition 3.28 Set ¶

Supports

depends_on definition 4.2 Binary operation ¶
- depends_on definition 4.3 Algebraic structure ¶
  - depends_on definition 4.6 Associativity ¶
    - depends_on definition 4.18 Semigroup ¶
      - depends_on definition 4.20 Monoid ¶
      - 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 definition 4.28 Distributivity ¶
    - depends_on definition 4.34 Algebra ¶
    - depends_on definition 4.31 Module ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 4.30 Ring ¶ ↺
  - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.34 Algebra ¶ ↺
    - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.30 Ring ¶ ↺
      - depends_on definition 4.33 Vector space ¶ ↺
      - depends_on proposition 4.68 Direct product of abelian groups ¶
      - depends_on proposition 4.52 $S_3$ is a smallest non-abelian group ¶
      - depends_on proposition 4.41 prop:alg-zn-group ¶
    - depends_on definition 4.18 Semigroup ¶ ↺
    - depends_on definition 4.12 Substructure ¶
    - 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 4.9 Neutral element ¶
    - depends_on definition 4.29 Action ¶
      - depends_on definition 4.31 Module ¶ ↺
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.21 Group ¶
      - depends_on proposition 4.17 Trivial kernel of a homomorphism ¶
    - depends_on definition 4.15 Kernel ¶
      - depends_on proposition 4.17 Trivial kernel of a homomorphism ¶ ↺
    - depends_on definition 4.20 Monoid ¶ ↺
    - depends_on proposition 4.10 Uniqueness of the neutral element ¶
- depends_on definition 4.4 Internal binary operation; magma ¶ ↺
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 example 4.78 The Euclidean group ¶
    - depends_on proposition 18.20 The Galilean group ¶
      - depends_on corollary 18.24 Galilean composition of velocities ¶
      - depends_on remark 18.22 Superseded, and in exactly what sense ¶
depends_on definition 4.73 External semidirect product ¶
- depends_on example 4.78 The Euclidean group ¶ ↺
- depends_on example 4.79 The Poincaré group ¶
- depends_on proposition 4.74 The semidirect product is a group ¶
  - depends_on proposition 4.75 The two factors inside the semidirect product ¶
    - depends_on example 4.79 The Poincaré group ¶ ↺
    - depends_on proposition 4.77 When a semidirect product is direct ¶ ↺
- depends_on proposition 18.20 The Galilean group ¶ ↺
- depends_on theorem 4.76 Internal characterization of the semidirect product ¶

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`	→	Set	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:53
`depends_on`	←	Binary operation	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:74
`depends_on`	←	Direct product of groups	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2393
`depends_on`	←	External semidirect product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:2692