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The basic algebraic structures with one binary operation are the following:
- A set with a binary operation.
- A set with an operation which is associative
- A semigroup with an identity element
- A monoid where each element has a corresponding inverse element
- A group with a commutative operation
The basic algebraic structures with two binary operations are the following:
- A set with two operations, often called addition and multiplication. The set with the operation of addition forms a commutative group, and with the operation of multiplication it forms a semigroup (many people define a ring so that the set with multiplication is actually a monoid). Addition and multiplication in a ring satisfy the distributive property
- A ring whose multiplication is commutative
- A commutative ring where the set with multiplication is a group.
- The whole numbers (natural numbers together with zero) with addition (or multiplication) is a monoid, but is not a group
- The integers with addition is a commutative group, but with multiplication is just a monoid
- The integers with addition and multiplication is a commutative ring, but not a field