Algebraic structure
Inmathematicsanalgebraic structureis asetwith one, two, or morebinary operationson it. The binary operation takes two elements of the set as inputs, and gives one element of the set as an output.
The basic algebraic structures withone binary operationare the following:
- A set with a binary operation.
- A set with an operation which isassociative
- A semigroup with anidentity element
- A monoid where each element has a correspondinginverse element
- A group with acommutative operation
The basic algebraic structures withtwo binary operationsare 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 thedistributive property
- A ring whose multiplication is commutative
- A commutative ring where the set with multiplication is a group.
Examples are
- The whole numbers (natural numberstogether with zero) with addition (or multiplication) is a monoid, but is not a group
- Theintegerswith 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
- Therational numbers,thereal numbersand thecomplex numberswith the ordinary addition and ordinary multiplication are fields.