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The set of real numbers has several standard structures: An order: each number is either less than or greater than any other number. Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a group and the pair of which together make it into a field.
For example, the set of natural numbers N is equipped with a natural pre-order structure, where ′ whenever we can find some other number so that + = ′. That is, n ′ {\displaystyle n'} is bigger than n {\displaystyle n} only because we can get to n ′ {\displaystyle n'} from n {\displaystyle n} using m {\displaystyle m} .
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
In mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include: They include: Cyclic orders , orderings in which triples of elements are either clockwise or counterclockwise
The long real line pastes together ℵ 1 * + ℵ 1 copies of the real line plus a single point (here ℵ 1 * denotes the reversed ordering of ℵ 1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ 1 in the long real line but not in the real ...
Mapping each natural number to the corresponding real number gives an example for an order embedding. The set complement on a powerset is an example of an antitone function. An important question is when two orders are "essentially equal", i.e. when they are the same up to renaming of elements.