Algebraic structures

https://en.wikipedia.org/wiki/List_of_group_theory_topics https://en.wikipedia.org/wiki/Glossary_of_group_theory

Algebraic structures

• Group-like

  Magma : closure

  Semigroup : magma + assoc

  Monoid : semigroup + identity

  Group : monoid + invertability

  Abelian group : group + commutativity

  Quasigroup : division always possible but need not be assoc

  Loop : quasigroup + identity

  Rack and quandle:

  Lie group

  Group theory

• Ring-like

  Ring

  Semiring

  Near-ring

  Commutative ring

  Integral domain

  Field

  Division ring

  Ring theory

• Lattice-like

  Lattice

  Semilattice

  Complemented lattice

  Total order

  Heyting algebra

  Boolean algebra

  Map of lattices

  Lattice theory

• Module-like

  Module

  Group with operators

  Vector space

  Linear algebra

• Algebra-like

  Algebra

  Associative

  Non-associative

  Composition algebra

  Lie algebra

  Graded

  Bialgebra

Algebraic structures related to groups

https://en.wikipedia.org/wiki/Category:Algebraic_structures

[Field](https://en.wikipedia.org/wiki/Field_(mathematics)) Finite field Galois theory Grothendieck group Group ring Group with operators [Heap](https://en.wikipedia.org/wiki/Heap_(mathematics)) Linear algebra [Magma](https://en.wikipedia.org/wiki/Magma_(algebra)) [Module](https://en.wikipedia.org/wiki/Module_(mathematics)) Monoid Monoid ring Quandle Quasigroup Quantum group [Ring](https://en.wikipedia.org/wiki/Ring_(mathematics)) Semigroup Vector space

Possible algebras

Possible algebras with these 5 axioms:

• closure/totality clo

• associativity assoc

• identity id

• invertability inv

• commutativity comm

5-choose-k = 5th row of Pascal's triangle: 1 5 10 10 5 1

• 1 × no axioms

• 1 × all five axioms

• 5 × single axiom

• 5 × axiom quadruples

• 10 × axiom pairs

• 10 × axiom triples

Known algebras

• 1 × all five axioms

  • ["cl","as","id","in","co"] : Abelian group

• 5 × single axiom

  • ["cl"] closure/totality : Magma

  • ["as"] associativity :

  • ["co"] identity :

  • ["id"] invertability :

  • ["in"] commutativity :

• 5 × axiom quadruples

  • ["cl","as","id","in"] : Group

  • ["cl","as","id","co"]

  • ["cl","as","in","co"]

  • ["cl","id","in","co"]

  • ["as","id","in","co"]

• 10 × axiom pairs

  • ["cl","as"] : Semigroup

  • ["cl","id"]

  • ["as","id"]

  • ["cl","in"]

  • ["as","in"]

  • ["id","in"]

  • ["cl","co"]

  • ["as","co"]

  • ["id","co"]

  • ["in","co"]

• 10 × axiom triples

  • ["cl","as","id"] : Monoid

  • ["cl","as","in"]

  • ["cl","id","in"]

  • ["as","id","in"]

  • ["cl","as","co"]

  • ["cl","id","co"]

  • ["as","id","co"]

  • ["cl","in","co"]

  • ["as","in","co"]

  • ["id","in","co"]