On the multiplicative structure of a class of ordered semirings (Q1821856)

The authors define a totally ordered (t. o.) semiring \((S,+,\cdot,\leq)\) in demanding that \((S,+,\leq)\) and (S,\(\cdot,\leq)\) are t. o. semigroups, a definition which excludes each t. o. ring. They pose the problem to describe all those semirings \((S,+,\cdot,\leq)\) in the case where \((S,+,\leq)\) is an Archimedean, naturally t. o. semigroup (an extensively investigated class of t. o. semigroups [cf. \textit{L. Fuchs}, Partially ordered algebraic systems (1963; Zbl 0137.020), Chap. XI, {\S} 2]). If \((S,+)\) is not cancellative, this problem is solved completely. If \((S,+)\) is cancellative, merely multiplications satisfying \(x\leq x^ 2\) for all \(x\in S\) are considered. Reviewer's remarks: The paper seems to deal with a more general case, since \((S,+,\leq)\) is always called an ''Archimedean, right naturally t. o. semigroup''. It is nowhere mentioned (but used) that such a semigroup is commutative and hence an Archimedean, naturally t. o. semigroup. This was stated in \textit{M. Satyanarayana} [Positively ordered semigroups (1979; Zbl 0411.06010), Thm. 3.21] (the proof given there is not complete [cf. \textit{H. J. Weinert}, Semigroup Forum 34, 235-242 (1986; Zbl 0605.06011)]). Moreover, common basic notions on t. o. semigroups are changed in this paper, though all given references (the two books cited above) use the common ones.