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| || colspan="5" | <small>**Each element of a [[Loop (algebra)|loop]] has a left and right inverse, but these need not coincide.</small>
|}
{| style="text-align:center"
|- align="center"
| style="border-bottom: 2px solid #303060" colspan=11| '''Ring-like structures'''
|-
! !! [[Magma]]* !! [[Semigroup]] !! [[Monoid]] !! [[Group]] !! [[Abelian Group]] !! [[Loop]] !! [[Quasigroup]] !! [[Groupoid]] !! [[Cathegory]] !! [[Semicathegory]]
|-
! [[Magma (algebra)|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 (mathematics)|Group]]
| Y || N || N || N || N || N || N || N || N || N
|-
! [[Abelian Group]]
| Y || N || N || N || N || N || N || N || N || N
|-
! [[Loop (algebra)|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 (mathematics)|Category]]
| Y || N || N || N || N || N || N || N || N || N
|-
! [[Semicategory]]
| Y || N || N || N || N || N || N || N || N || N
|}
