definition 4.2 Binary operation

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:65 · p. 57

ground object -- no derivation owed

Rests on

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 3.1 Logical proposition ¶
    - depends_on definition 3.38 Ordered pair and Cartesian product ¶
- depends_on definition 3.28 Set ¶ ↺

Supports

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 4.32 Field ¶
      - depends_on definition 4.21 Group ¶
    - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.32 Field ¶ ↺
      - depends_on definition 4.31 Module ¶
    - 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 4.28 Distributivity ¶
  - depends_on definition 4.34 Algebra ¶
  - depends_on definition 4.31 Module ¶ ↺
  - 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 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 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 4.12 Substructure ¶
  - depends_on definition 5.121 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 ¶ ↺

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`	→	Cartesian product	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:74
`depends_on`	→	Map	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:74
`depends_on`	←	Algebraic structure	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:82
`depends_on`	←	Internal binary operation; magma	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:124