Orderings on semirings

Abstract

Though more general definitions are sometimes used, for this paper a semiring will be defined to be a set S with two operations + (addition) and �9 (multiplication); with respect to addition, S is a commutative monoid with 0 as its identity dement�9 With respect to multiplication, S is a (generaily noncommutative) monoid with 1 as its identity element. Connecting the two algebraic structures are the distributive laws, a.(b+ c) = a.b+ a.c and (a+ b).c = a.c+ b.c for all a,b,c E S, and the requirement a.0 = 0.a = 0 for every a C S. The reader is referred to [3] and [4] for an introduction to the theory of semirings. Semirings have proven to be useful in theoretical computer science, in particular for studying automata and formal languages. Several authors have considered total order relations on semirings (e.g. [8, 9]). In this paper, we begin by looking at special classes of semirings which arise naturally in the theory of formally real (not necessarily commutative) fields and in the study of (commutative) real algebraic geometry�9 In the second section of the paper, we give characterizations of the semirings which arise in this way. Using this work as motivation, the third section of the paper suggests a new notion of ordering for commutative semirings which generalizes the concepts found in real algebraic geometry. In the most general case, these orderings turn out to have a somewhat weaker connection with total order relations than in the case of rings.