Assign Ordered Field For Free

A set F together with two operations + and ? and a relation < satisfying the 13 axioms above is called an ordered field. Thus the real numbers are an example of an ordered field. Another example of an ordered field is the set of rational numbers Q with the familiar operations and order.

Rational numbers are an ordered field The integers do not form a field! They almost do though, but just don't have multiplicative inverses (except that the integer 1 is its own multiplicative inverse ? it is also the multiplicative identity).

The answer must be (b) the rational numbers. The rational numbers are a sub-field of the real numbers, and inherit the order of the latter, most basic example of an ordered field.

Ordered set, we can have x ~ y . An "ordered field" however, is usually taken to be a totally ordered field -- so without extra conditions, x~y => x=y . "Partially ordered fields" have been studied, but you have to specify what conditions you mean.

An ordered set is a set S in which an order is defined. He also defined order in his book: Let S be a set. An order on S is a relation, denoted by <, with the following two properties: If x, y ? S then one and only one of x < y, x = y, x > y is true.

Definition of an ordered field: An ordered field is a field containing a subset of elements closed under addition and multiplication and having the property that every element in the field is either 0, in the subset, or has its additive inverse in the subset.

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field is isomorphic to the reals.