**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.

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