Quasi-ordered fields. (English) Zbl 0629.12022

The author calls a field with a binary relation \(\leq\) “quasi-ordered” if a set of axioms closely related to the axioms of ordered fields is fulfilled. Every ordered field is quasi-ordered, and on every Krull valued field a quasi-order is induced by the valuation, so this definition is a common generalization of both orderings and valuations.
On the other hand these are the only possible types of quasi-ordered fields, since the main theorem says that every such field either is an ordered field or a Krull valued field whose valuation induces the given quasi-order.


12J15 Ordered fields
12J10 Valued fields
Full Text: DOI


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