Template:Group-like structures

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Requirements of group-like structures
Totality Identity Associativity Inverses Commutativity
Magma Yes No No No No
Unital Yes Yes No No No
Quasigroup Yes No No Yes No
Loop Yes Yes No Yes No
Semigroup Yes No Yes No No
Monoid Yes Yes Yes No No
Group Yes Yes Yes Yes No
Abelian Group Yes Yes Yes Yes Yes
Groupoid No Yes Yes Yes No
Category No Yes Yes No No
Semicategory No No Yes No No
Note: A quasigroup with associativity (equivalently, a semigroup with
inverses) already has an identity element, and is therefore a group.

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