1.A semigroup has an associative binary operation, but might not have an identity element.
2.A monoid is a semigroup which does have an identity but might not have an inverse for every element.
3.A quasigroup satisfies a requirement that any element can be turned into any other by a unique pre- or post-operation; however the binary operation might not be associative. All groups are monoids, and all monoids are semigroups.
4.Rings and fields—structures of a set with two particular binary operations, (+) and (×) ring and field Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied.
The most important of these are rings, and fields. Distributivity generalised the distributive law for numbers, and specifies the order in which the operators should be applied, (called the precedence).
- For the integers (a + b) × c = a×c+ b×c and c × (a + b) = c×a + c×b, and × is said to be distributive over +. A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an Abelian group.
- Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not allowed. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.
- The integers are an example of a ring. The integers have additional properties which make it an integral domain. A field is a ring with the additional property that all the elements excluding 0 form an Abelian group under ×.
- The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a-1.
- The rational numbers, the real numbers and the complex numbers are all examples of fields.