Orderings of Semifields

Abstract

A fundamental characterization of orderable elds is given by Artin and Schreier: A eld F can be ordered if and only if it is formally real. Knebusch, Rosenberg and Ware proved that the space of orderings of F can be topologized to make a Boolean space (compact, Hausdor¤, and totally disconnected) providing a connection between the orderings of a eld and the theory of quadratic forms. A semi eld is an algebraic structure which is a multiplicative abelian group and an additive commutative semigroup at the same time, and multiplication is distributive with respect to addition. In this paper, we study the orderings of general semi elds, with particular attention on idempotent semi elds (such as are used in tropical algebra) and subsemi elds of formally real elds. We show that some subsemi elds completely recapture the ordering structure of a eld, while others can have dramatically di¤erent orderings. Interestingly though, it turns out that the space of orderings of a semi eld is also a Boolean space.