Algebraic structure

set in category theory with one or more finitary operations defined on it

In mathematics an algebraic structure is a set with one, two or more binary operations on[needs to be explained] it.

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.

Examples are

  • 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

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