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

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