Archimedean property of ordered semirings

Abstract

In this paper we are concerned with the problem of determining when the multiplicative semigroup is o-Archimedean if the additive semigroup is o-Archimedean and conversely, in totally ordered semirings (t.o.s.r.). This is in continuation of the author's work in [i] and [2] on the problems how far the properties of multiplicative structure are reflected in the additive structure and vice-versa in ordered semirings. A semiring (S,+,.) is said to be a totally ordered semiring if the additive semigroup (S,+) and the multiplicative semigroup (S,') are totally ordered (t.o.) semigroups under the same total order relation. For any t.o. semigroup, say (S,'), an element x in S is said to be non-negative (non-positive) if x2 ~x (x 2 ~ x); x is said to be positively ordered (negatively ordered) in strict sense if xy ~ x and xy ~ y (xy < x and q m