# 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:&#x20;

  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](https://en.wikipedia.org/wiki/Finite_field) [Galois theory](https://en.wikipedia.org/wiki/Galois_theory) [Grothendieck group](https://en.wikipedia.org/wiki/Grothendieck_group) [Group ring](https://en.wikipedia.org/wiki/Group_ring) [Group with operators](https://en.wikipedia.org/wiki/Group_with_operators) \[Heap]\(<https://en.wikipedia.org/wiki/Heap_(mathematics>)) [Linear algebra](https://en.wikipedia.org/wiki/Linear_algebra) \[Magma]\(<https://en.wikipedia.org/wiki/Magma_(algebra>)) \[Module]\(<https://en.wikipedia.org/wiki/Module_(mathematics>)) [Monoid](https://en.wikipedia.org/wiki/Monoid) [Monoid ring](https://en.wikipedia.org/wiki/Monoid_ring) [Quandle](https://en.wikipedia.org/wiki/Quandle) [Quasigroup](https://en.wikipedia.org/wiki/Quasigroup) [Quantum group](https://en.wikipedia.org/wiki/Quantum_group) \[Ring]\(<https://en.wikipedia.org/wiki/Ring_(mathematics>)) [Semigroup](https://en.wikipedia.org/wiki/Semigroup) [Vector space](https://en.wikipedia.org/wiki/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"]
