# Algebraic structures

[https://en.wikipedia.org/wiki/List\_of\_group\_theory\_topics](https://en.wikipedia.org/wiki/List_of_group_theory_topics) [https://en.wikipedia.org/wiki/Glossary\_of\_group\_theory](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](https://en.wikipedia.org/wiki/Category:Algebraic_structures)

\[Field]\([https://en.wikipedia.org/wiki/Field\_(mathematics](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](https://en.wikipedia.org/wiki/Heap_\(mathematics))) [Linear algebra](https://en.wikipedia.org/wiki/Linear_algebra) \[Magma]\([https://en.wikipedia.org/wiki/Magma\_(algebra](https://en.wikipedia.org/wiki/Magma_\(algebra))) \[Module]\([https://en.wikipedia.org/wiki/Module\_(mathematics](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](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"]
