Участник:Dgerasimov/Algebra

Материал из Викиконспекты
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Taken from [1]

Group-like structures
Totality* Associativity Identity Inverses Commutativity
Magma Y N N N N
Semigroup Y Y N N N
Monoid Y Y Y N N
Commutative monoid Y Y Y N Y
Group Y Y Y Y N
Abelian Group Y Y Y Y Y
Loop Y N Y Y** N
Quasigroup Y N N N N
Groupoid N Y Y Y N
Category N Y Y N N
Semicategory N Y N N N
*Closure, which is used in many sources to define group-like structures, is an equivalent axiom to totality, though defined differently.
**Each element of a loop has a left and right inverse, but these need not coincide.


Ring-like structures
Magma* Semigroup Monoid Group Abelian Group Loop Quasigroup Groupoid Cathegory Semicathegory
Magma Y N N N N N N N N N
Semigroup Y N N N N N N N N N
Monoid Y N N N N N N N N N
Group Y N N N N N N N N N
Abelian Group  ? Ring Unital ring  ? Commutative ring  ?  ?  ?  ?  ?
Loop Y N N N N N N N N N
Quasigroup Y N N N N N N N N N
Groupoid Y N N N N N N N N N
Category Y N N N N N N N N N
Semicategory Y N N N N N N N N N