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Set Closed binary operation Multiplicative operation (•) that Neighborhood distributes over additive (+) Ringoid Associativity Magma -Inverses Ring-like associativity + associativity and identity (0) Structures 0-x=x·0=0 (D, ·) is a semigroup Semigroup Idempotence- Quasigroup -Idempotence Band Rng Semi-rng +commutes "Quasi-rng" Pique Additive inverse Identity Identity (D, +) is a monoid Group-like Multiplicative identity (1) Structures Left Bol Identity. Left Bol Loop Multiplicative identity (1) Multiplicative identity (1) Monoid Loop Moufang Union (D, ·) is a monoid of axioms Loop Ring Semi-ring "Quasi-ring" -Inverses- Associativity Topological Right Bol ldentity— Right Bol Loop Additive inverse- + commutes Space (R, +) is Abelian- -Distance function Group Zero product property Commutability -Union of axioms . 0 ≠1 ·commutes Boolean Ring Metric Space 0 ≠1 Additive inverse Multiplicative Domain Commutative Topological -Idempotence Abelian Group Group Compléteness inverse Ring Zero product property Differentiability Complete Metric ·commutes 0≠1 Multiplicative Near Ring Space inverse 0≠1 Lie Group . Integral Division Ring Multiplicative Field Domain inverse (D, +) and (D, ·) are Abelian Groups commutes

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