Ordered differential fields (Q5906426)

Note that a nonzero derivation on a linearly ordered field cannot be a monotone operator. So the classical definition of an order for a differential field makes no sense, since it imposes the condition on the derivation to be a monotone operator. The author introduces the notion of a partially ordered differential field as a partially ordered field with some derivation that is monotone on some convex subset of the field. A set \(M\) is called convex if it contains at least two elements and \((\forall x,y,z)(x,z\in M \;\& \;x\leq y\leq z\Rightarrow y\in M)\). It is easy to see that in an ordered and differential field \(G\) there exists a convex set \(U\) such that \(0\in U\) and the derivation is monotone on \(U\). A set of consistent elements of the field \(G\) is a maximal set with these properties. The author considers only linearly ordered fields. For a linearly ordered field, the set of consistent elements is unique and the author considers the case in which this set is maximal in some proper sense and the derivation is monotone on this set. Such a class of fields is called a class of ordered differential fields. In the first part of the article the author gives some examples of such fields. Besides the field of rational functions, a Hardy field is also presented as an example. In the second part the author proves some arithmetical properties of elements of ordered differential fields. In the third part he proves that an ordered differential field with interval topology is a totally disconnected topological space. In particular, the derivation is continuous. Here the author proves theorems about continuation of the derivation on an ordered field and about completion of an ordered differential field. In the last section the author gives some theorems on continuation of the order and derivation onto some algebraic and some transcendental extensions of an ordered differential field.