Template:Group-like structures

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Revision as of 15:53, 20 August 2012 by imported>LokiClock (Show unital magma after magma)
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Requirements of group-like structures
Totality Associativity Identity Inverses Commutativity
Magma Yes No No No No
Unital Yes No Yes No No
Quasigroup Yes No No Yes No
Loop Yes No Yes Yes No
Semigroup Yes Yes No 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 Yes No 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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