Ordered Rings and Fields

Abstract

Summary.We introduce ordered rings and fields following Artin-Schreier's approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways [8,5,4,9]. This is the continuation of the development of algebraic hierarchy in Mizar [2,3]. MSC: 12J15 03B35Keywords: commutative algebra; ordered fields; positive cones MML identifier: REALALG1, version: 8.1.05 5.40.1289 On Order RelationsLet X be a set and R be a binary relation on X. We say that R is strongly reflexive if and only if (Def. 1) R is reflexive in X.We say that R is totally connected if and only if (Def. 2) R is strongly connected in X.One can check that there exists a binary relation on X which is strongly reflexive and there exists a binary relation on X which is totally connected and every binary relation on X which is strongly reflexive is also reflexive and every binary relation on X which is totally connected is also strongly connected.Let X be a non empty set. One can check that every binary relation on X which is strongly reflexive is also non empty and every binary relation on X which is totally connected is also non empty. Now we state the propositions: