ordered field

Noun

Meaning

ordered field (plural ordered fields)

(algebra) A field which has an order relation satisfying these properties: trichotomy, transitivity, preservation of an inequality when the same element is added to both sides, and preservation of an inequality when the same strictly positive element is multiplied to both sides.

Examples

Any Dedekind-complete ordered field is isomorphic to the real numbers. Source: Internet

Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. Source: Internet

Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Source: Internet

Every subfield of an ordered field is also an ordered field in the inherited order. Source: Internet

For example, it is not enough to construct an ordered field with infinitesimals. Source: Internet

Lam (2005) p. 41 Lam (2005) p. 232 Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. Source: Internet

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