On the additive and multiplicative structures of certain classes of ordered semirings (Q1312270)

A totally ordered (t.o.) semigroup \((S,\cdot)\) is positively ordered in the strict sense if \(xy\geq x\) and \(xy\geq y\) for all \(x\) and \(y\) in \(S\). \((S,\cdot)\) is right naturally totally ordered (r.n.t.o.) if \((S,\cdot)\) is positively ordered in the strict sense and if \(a< b\), then \(b= at\) for some \(t\) in \(S\). \((S,\cdot)\) is \(O\)-Archimedean if for every \(x\) and \(y\) in \(S\), one of the following is satisfied: \(x\leq y\leq x^ m\); \(y^ m\leq x\leq y\); \(y\leq x\leq y^ m\); \(x^ m\leq y\leq x\) for some natural number \(m\). A semiring \((S,+,\cdot)\) is said to be a totally ordered semiring if the additive semigroup \((S,+)\) and the multiplicative semigroup \((S,\cdot)\) are t.o. semigroups under the same total order relation. \(E(+)\) denotes the set of all additive idempotents in \((S,+)\). The properties of the t.o. semirings \((S,+,\cdot)\), in which \((S,\cdot)\) is r.n.t.o. and \(O\)-Archimedean, are studied when (i) \(| E(+)|= 1\) and \((S,\cdot)\) is non-cancellative, (ii) \(| E(+)|= 0\) and \((S,\cdot)\) is cancellative. The structure of t.o. semirings \((S,+,\cdot)\) in which \((S,+)\) is r.n.t.o. and \(O\)-Archimedean is studied, when (i) \((S,+)\) is non- cancellative, (ii) \((S,+)\) is cancellative and \(S\) has no minimal element. The purpose of this paper is to fill some of the gaps that are left in two papers by \textit{M. Satyanarayana}, the first author and \textit{D. Umamaheswara Reddy} [Semigroup Forum 33, 251-255 (1986; Zbl 0582.06019); ibid. 35, 175-180 (1987; Zbl 0617.16027)].