Some Conditions on Semirings

Abstract

We usually consider semirings on which some sort of additional conditions have been imposed. Many such conditions were given in Chapter 1 and examples given of semirings which do or do not satisfy them. We now want to consider consequences of imposing some of these conditions on a semiring. In particular we will first look at the condition of being an additively-idempotent semiring and at the stronger condition of being a simple semiring. Then we will consider some weaker versions of the condition that elements have additive or multiplicative inverses. Finally, we will take up a condition which guarantees the existence of “enough” multiplicative units.