Difference between revisions of "Template:Group-like structures"
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(adding Commutative monoid) |
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| colspan="6" | <small>{{note|Alpha|α}}[[Closure (mathematics)|Closure]], which is used in many sources, is an equivalent axiom to totality, though defined differently.</small> | | colspan="6" | <small>{{note|Alpha|α}}[[Closure (mathematics)|Closure]], which is used in many sources, is an equivalent axiom to totality, though defined differently.</small> | ||
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Latest revision as of 21:08, 28 July 2020
Group-like structures | |||||
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Totalityα | Associativity | Identity | Invertibility | Commutativity | |
Semigroupoid | Unneeded | Required | Unneeded | Unneeded | Unneeded |
Small Category | Unneeded | Required | Required | Unneeded | Unneeded |
Groupoid | Unneeded | Required | Required | Required | Unneeded |
Magma | Required | Unneeded | Unneeded | Unneeded | Unneeded |
Quasigroup | Required | Unneeded | Unneeded | Required | Unneeded |
Unital Magma | Required | Unneeded | Required | Unneeded | Unneeded |
Loop | Required | Unneeded | Required | Required | Unneeded |
Semigroup | Required | Required | Unneeded | Unneeded | Unneeded |
Inverse Semigroup | Required | Required | Unneeded | Required | Unneeded |
Monoid | Required | Required | Required | Unneeded | Unneeded |
Commutative monoid | Required | Required | Required | Unneeded | Required |
Group | Required | Required | Required | Required | Unneeded |
Abelian group | Required | Required | Required | Required | Required |
^α Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently. |