On <Emphasis Type="Italic">N</Emphasis>-Summations,II

Given any R-semimodule M equipped with a semitopology we construct an N-protosummation for M. If satisfies certain properties, then a similar construction leads to an unconditional N-summation for M, that is an N-summation for M equipped with the trivial prenorm MD over the N-summation (D^N,_D) for D. Conversely any N-protosummation on M gives rise to a topology . If both and satisfy a certain separation property, then and form a Galois connection. Dedicated to my friend and collegue Nico Pumplün on the occasion of his 70th birthdayMathematics Subject Classifications (2000) 16Y60, 54A05.     

On V-semirings and Semirings All of Whose Cyclic Semimodules Are Injective

In this article, we introduce and study V- and CI-semirings—semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties of V-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the bounded distributive lattices, Gelfand, subtractive, semisimple, and antibounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple antibounded CI-semirings which solve two earlier open problems for these classes of CI-semirings.     

Monoid Valuations and Value Ordered Supervaluations

We complement two papers on supertropical valuation theory ([11 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 . Preprint at arXiv:1003.1101 .[Crossref], [Web of Science ®] , [Google Scholar]], [12 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2013 ). Dominance and transmissions in supertropical valuation theory . Commun. Algebra 41 ( 7 ): 2736 – 2782 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]) by providing natural examples of m-valuations (= monoid valuations), and afterwards of supervaluations and transmissions between them. These supervaluations have values in totally ordered supertropical semirings, and the transmissions discussed respect the orderings. We develop the basics of the theory of such semirings and transmissions.     

模糊关系下的半环的广义模糊双理想

给出了半环的广义模糊双理想的概念并讨论了模糊关系下的半环的广义模糊双理想的一些代数性质.     

半环的星结构

引入了半环的星元素和星理想等概念,并且讨论了半环的星结构,得到了半环的星结构定理.     

The Semigroup Structure of Left Clifford Semirings

In this paper,we generalize Clifford semirings to left Clifford semirings by means of the so-called band semirings.We also discuss a special case of this kind of semirings,that is, strong distributive lattices of left rings.     

On N-Summations, I

This paper continues recent work on N-summations (see [8, 9, 12]). More specifically, it addresses the issue of existence of N-summations both for cone semirings and for prenormed semitopological semimodules. In the case of a cone semiring C we assume N-order completeness plus compatibility of N-joins with addition and multiplication to make the class of summarily bounded elements of C ^N into an N-summation for C. In the case of a prenormed semitopological semimodule M we use certain completeness properties of semitopologies on M to make the class of Cauchy elements of M ^N into an N-summation for M. Results on semitopologies and their connection with closure operators are contained in the Appendix.