Author
Jonathan S. Golan
Bio: Jonathan S. Golan is an academic researcher from University of Haifa. The author has contributed to research in topics: Semiring & Ring theory. The author has an hindex of 5, co-authored 8 publications receiving 1104 citations.
Topics: Semiring, Ring theory, Kleene algebra, Maximal ideal, Idempotence
Papers
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Book•
14 Mar 2014
TL;DR: In this paper, the authors define Hemirings and semirings as "sets and relations with values with values in a semiring" and define a set of conditions on semimodal construction.
Abstract: Preface. 1. Hemirings and semirings: definitions and examples. 2. Sets and relations with values in a semiring. 3. Building new semirings from old. 4. Some conditions on semirings. 5. Complemented elements in semirings. 6. Ideals in semirings. 7. Prime and semiprime ideals in semirings. 8. Factor semirings. 9. Morphisms of semirings. 10. Kernels of morphisms. 11. Semirings of fractions. 12. Euclidean semirings. 13. Additively-regular semirings. 14. Semimodules over semirings. 15. Factor semimodules. 16. Some constructions for semimodules. 17. Free, projective, and injective semimodules. 18. Localization of semimodules. 19. Linear algebra over a semiring. 20. Partially-ordered semirings. 21. Lattice-ordered semirings. 22. Complete semirings. 23. Complete semimodules. 24. CLO-semirings. 25. Fixed points of affine maps. References. Index of applications. Index of terminology.
977 citations
Book•
30 Jun 2003
TL;DR: In this article, the authors introduce partially ordered and partially ordered semirings, which are partially ordered Semimodels, as well as partially ordered Matrix Semirings (Semimodules).
Abstract: Preface. Introduction. 1: Semirings. 2: Partially-Ordered Semirings. 3: Complete Semirings. 4: Residuated Semirings. 5: Matrix Semirings. 6: Symmetric Extension of a Semiring. 7: Semimodules. 8: Homomorphisms between Semimodules. 9: Affine Maps between Semimodules. 10: Partially-ordered Semimodules. 11: Eigenelements. 12: Permanents and Determinants. Bibliography. Index of Applications. Index of Terminology.
161 citations
Book•
09 Dec 2010
TL;DR: In this paper, the authors provide motivating examples of semiring-valued subsemigroups and submonoids, as well as semiringvalued submodules and subspaces.
Abstract: Preface. Some (hopefully) motivating examples. 0. Background material. 1. Powers of a semiring. 2. Relations with values in a semiring. 3. Change of base semirings. 4. Convolutions. 5. Semiring-valued subsemigroups and submonoids. 6. Semiring-valued groups. 7. Semiring-valued submodules and subspaces. 8. Semiring-ideals in semirings and rings. References. Index.
27 citations
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TL;DR: In this article, it was shown that the family of all ideals of a semiring is, in a natural way, an ideal semiring, and the definitions of semiring are the same as in this paper.
Abstract: Ideals play a fundamental role in ring theory and it is therefore natural to consider them also in the context of semiring theory. Here their role is no less important, though we will often have to restrict our consideration to special types of ideals. In particular, we will show that, as in the case of rings, the family of all ideals of a semiring is, in a natural way, a semiring. Formally, the definitions in the two situations are the same.
9 citations
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TL;DR: This paper initiates the study of soft semirings by using the soft set theory, and the notions of soft Semirings, soft subsemirings,soft ideals, idealistic softSemirings and soft semiring homomorphisms are introduced, and several related properties are investigated.
Abstract: Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we initiate the study of soft semirings by using the soft set theory. The notions of soft semirings, soft subsemirings, soft ideals, idealistic soft semirings and soft semiring homomorphisms are introduced, and several related properties are investigated.
579 citations
TL;DR: In this article, a brief introduction to idempotent and tropical mathematics is given, which can be seen as the result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant tends to zero taking imaginary values.
Abstract: This paper is a brief introduction to idempotent and tropical mathematics. Tropical mathematics can be treated as the result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant ℏ tends to zero taking imaginary values. Bibliography: 187 titles.
242 citations
TL;DR: A weighted version of MSO logic is introduced and it is proved that, for commutative semirings, the behaviours of weighted automata are precisely the formal power series definable with particular sentences of the authors' weighted logic.
240 citations
TL;DR: In this paper, an algebraic approach to idempotent functional analysis is presented, which is an abstract version of the traditional functional analysis developed by V. P. Maslov and his collaborators.
Abstract: This paper is devoted to Idempotent Functional Analysis, which is an “abstract” version of Idempotent Analysis developed by V. P. Maslov and his collaborators. We give a brief survey of the basic ideas of Idempotent Analysis. The correspondence between concepts and theorems of traditional Functional Analysis and its idempotent version is discussed in the spirit of N. Bohr's correspondence principle in quantum theory. We present an algebraic approach to Idempotent Functional Analysis. Basic notions and results are formulated in algebraic terms; the essential point is that the operation of idempotent addition can be defined for arbitrary infinite sets of summands. We study idempotent analogs of the basic principles of linear functional analysis and results on the general form of a linear functional and scalar products in idempotent spaces.
222 citations
03 Oct 2005
TL;DR: This thesis proposes an analytical model for evaluating timetable stability and robustness based on max-plus algebra which has been implemented in the software PETER (Performance Evaluation of Timed Events in Railways) which enables users to analyse large-scale network timetables in real-time.
Abstract: Reliability of railway operations becomes more and more demanding with increasing train traffic, which asks for stable and robust timetables capable of neutralizing deviations from scheduled time-distance paths and stabilizing delay propagation. Timetable performance evaluation is thus a crucial aspect in the railway timetable design process to guarantee and maintain reliability of operations. Feedback from realized railway operations is essential to evaluate the performance of the timetable in practice and to find and improve structural shortcomings in the timetable design. This thesis presents the developed software TNV-Prepare which recovers the infrastructure utilization of train traffic from train detection data of the safety and signalling systems based on daily records of the train describer systems, the so-called TNV-logfiles. A case-study at the Dutch railway station Eindhoven demonstrates the potential of a statistical analysis of this train traffic data to identify structural sources of delays. Railway operations are characterized by many network interdependencies resulting from the timetable and shared usage of railway infrastructure. Therefore, a railway timetable must be carefully tested on stability and robustness before implementation using a mathematical model of the scheduled railway operations. This thesis proposes an analytical model for evaluating timetable stability and robustness based on max-plus algebra. The max-plus linear system approach gives a formal stability test based on a max-plus eigenvalue problem, identifies critical events and processes, quantifies robustness in terms of recovery times, and computes the propagation of delays over time and space. The method has been implemented in the software PETER (Performance Evaluation of Timed Events in Railways) which enables users to analyse large-scale network timetables in real-time. A case-study of the Dutch national railway timetable illustrates the developed methodology.
183 citations