What is Algebraic structure?
In [[mathematics]] an '''algebraic structure''' is a [[set]] with one, two, or more [[binary operation]]s on 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 with '''one binary operation''' are the following: *[[Magma (mathematics)]] :: A set with a binary operation. *[[Semigroup]] :: A set with an operation which is [[associative]] *[[Monoid]] :: A semigroup with an [[identity element]] *[[Group (mathematics)|Group]] :: A monoid where each element has a corresponding [[inverse element]] *[[Commutative group]] :: A group with a [[commutative|commutative operation]] The basic algebraic structures with '''two binary operations''' are the following: *[[Ring (mathematics)|Ring]] :: 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]] *[[Commutative ring]] :: A ring whose multiplication is commutative *[[Field (mathematics)|Field]] :: A commutative ring where the set with multiplication is a group. Examples are *The whole numbers ([[natural number]]s together with zero) with addition (or multiplication) is a monoid, but is not a group *The [[integer]]s 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 *The [[rational number]]s, the [[real number]]s and the [[complex number]]s with the ordinary addition and ordinary multiplication are fields. == Related pages == *[[Abstract algebra]] [[Category:Abstract algebra]]