definition 4.3 Algebraic structure

open in the book · parts/02-mathematical-methods/02-algebraic-structures.tex:77 · p. 57

ground object -- no derivation owed

Rests on

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

Supports

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 4.33 Vector space ¶
      - depends_on lemma A.44 $\R$ is an ordered field ¶
      - depends_on theorem A.35 Completeness of $\R$ ¶
    - depends_on definition 4.21 Group ¶
      - depends_on definition 4.37 Cayley table ¶
      - depends_on definition 4.65 Direct product of groups ¶
      - depends_on definition 4.35 The group axioms, restated one at a time ¶
      - depends_on definition 4.43 Group homomorphism ¶
      - depends_on definition 4.31 Module ¶
      - depends_on definition 4.49 Permutation group ¶
      - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.36 Subgroup ¶
      - depends_on definition A.561 Coadjoint action and equivariance ¶
      - 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 ¶
      - … 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 4.28 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 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 5.128 Derivation ¶
    - depends_on remark 22.31 The bracket is a Lie algebra structure ¶
      - depends_on remark 22.39 Poisson's theorem does not manufacture new constants ¶
  - depends_on definition 5.124 Ideal ¶
    - depends_on proposition 5.164 The kernel is an ideal ¶
      - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶
  - depends_on definition 5.127 Lie algebra ¶
    - depends_on definition 24.40 Lie–Poisson bracket ¶
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
      - depends_on remark 29.27 Euler's equations as a Lie–Poisson system ¶
    - depends_on proposition 5.167 Adjoint representation ¶
      - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
    - depends_on proposition 5.139 Dimension of the symplectic group ¶
      - depends_on proposition 24.5 Properties of the symplectic group ¶
      - depends_on remark 22.21 The count ¶
    - depends_on remark 22.31 The bracket is a Lie algebra structure ¶ ↺
  - depends_on definition 5.122 Linear hull ¶
    - depends_on definition 5.124 Ideal ¶ ↺
    - depends_on definition 5.123 Subalgebra ¶
  - depends_on definition 5.123 Subalgebra ¶ ↺
  - depends_on definition 5.162 Representation of an algebra ¶
    - depends_on definition 5.163 Faithful, equivalent, invariant ¶
      - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
      - depends_on proposition 5.164 The kernel is an ideal ¶ ↺
    - depends_on proposition 5.167 Adjoint representation ¶ ↺
    - depends_on proposition 5.164 The kernel is an ideal ¶ ↺
  - depends_on proposition 5.125 prop:lin-matrix-algebra ¶
    - depends_on corollary 5.168 cor:rep-adjoint-faithful ¶ ↺
    - depends_on definition 5.162 Representation of an algebra ¶ ↺
    - depends_on definition 9.21 Matrix exponential ¶
      - depends_on proposition 9.22 The exponential and its derivative ¶
    - depends_on proposition 5.139 Dimension of the symplectic group ¶ ↺
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 ¶

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`	→	Binary operation	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:82
`depends_on`	←	Associativity	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:146
`depends_on`	←	Distributivity	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:723
`depends_on`	←	Internal binary operation; magma	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:124
`depends_on`	←	Neutral element	declared	parts/02-mathematical-methods/02-algebraic-structures.tex:187