Algebraic structure

From Simple English Wikipedia, the free encyclopedia

In mathematics an algebraic structure is a set with one, two or more binary operations on it.

With one binary operation these are the basic ones

  • Semigroup
It is a set with an operation which is associative
  • Monoid
It is a semigroup with neutral element
It is a monoid where each element has its inverse
  • Commutative Group
It is group with a commutative operation

With two binary operations are:

The operation are called the sum and the product genericaly. With the sum forms a commutative group, and with the product it is a semigroup. There is distributivity between the product and the sum
It is a ring where the product is commutative. There is distributivity and all elements different from the sum's neutral element have inverse

Examples are

  • The natural numbers with the sum is a semigroup and also a monoid. It is not a group
  • The integer numbers with the sum and multiplication is a ring, but not a field