Algebraic structure
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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
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- It is a set with an operation which is associative
- Monoid
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- It is a semigroup with neutral element
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- It is a monoid where each element has its inverse
- Commutative Group
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- It is group with a commutative operation
With two binary operations are:
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- 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
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- 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 is group and also a commutative group
- The integer numbers with the sum and multiplication is a ring, but not a field
- The rational numbers, the real numbers and the complex numbers with the ordinary sum and ordinary multiplication are fields.