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! [[Monoid]]
| Y || Y || Y || N || N
|-
! [[Monoid|Commutative monoid]]
| Y || Y || Y || N || Y
|-
! [[Group (mathematics)|Group]]
|-
! [[Abelian Group]]
| Y ? || N [http://en.wikipedia.org/wiki/Ring_%28mathematics%29 Ring] || N [http://en.wikipedia.org/wiki/Ring_%28mathematics%29 Unital ring] || N ? || N [http://en.wikipedia.org/wiki/Commutative_ring Commutative ring] || N ? || N ? || N ? || N ? || N?
|-
! [[Loop (algebra)|Loop]]
