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Associativity and commutativity together imply that the order that addition is performed is irrelevant. An algebra satisfying only associativity is called a semigroup, while a semigroup that also satisfies commutativity is called a commutative semigroup or an Abelian semigroup.
When other axioms are added for zero and negation, then the algebra is called a group, examples given on online math forum; and when commutative, an Abelian group. Groups are some of the most important algebraic structures in modern mathematics. read more on math forum.
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