# Surveying Lattice-Ordered Fields

• R. H. Redfield
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 7)

## Abstract

The object of what follows is to give a brief overview of the theory of lattice-ordered fields. While I have included no proofs, I have tried to give ample references for anyone interested in seeing the details. Section 1 briefly sketches the history behind the subject and section 2 recalls some basic definitions. The remainder reviews what is presently known: section 3 describes methods of constructing lattice-ordered fields; section 4 concerns the maximal totally ordered subfield; section 5 considers lattice-ordered fields as vector lattices; section 6 describes representations of lattice-ordered fields by means of power series fields; section 7 discusses the number of different compatible lattice orderings; section 8 deals with extensions to total orders; section 9 investigates the lattice-ordering of simple algebraic extensions of totally ordered fields; and section 10 lists some open questions. Of course not all results are mentioned below. I have chosen those that seem to me to be the most fundamental and the most interesting.

I have cited specific results in one of two ways: if an author has given the n th result in the m th section the number m.n, then I have used that number; otherwise, I have used the number of the page on which the result occurs.

## Keywords

Vector Lattice Total Order Convex Subgroup Positive Multiplicative Inverse Supporting Collection
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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