On the additive and multiplicative structures of certain classes of ordered semirings
type: Item
instance of: scholarly article
title: On the additive and multiplicative structures of certain classes of ordered semirings
zbMATH Open document ID: 0791.06015
author: J. Hanumanthachari
author: Thatimakula Vasanthi
published in: Southeast Asian Bulletin of Mathematics
publication date: 1994-07-07T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 16Y60
zbMATH DE Number: 493266
zbMATH Keywords: totally ordered semigroups
zbMATH Keywords: right naturally totally ordered
zbMATH Keywords: positively ordered in the strict sense
zbMATH Keywords: \(O\)-Archimedean
zbMATH Keywords: totally ordered semiring
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 493266