Generalized semialgebras over semirings

  • Hanns Joachim Weinert
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1320)


For an associative ring S, the classical concept of an algebra A over S forces S to be commutative in nearly all cases of some interest. E. g., polynomial rings or matrix rings over a non-commutative ring S with identity are not classical algebras over S. For this reason, H. Zassenhaus [21] and G. Pickert [13] have introduced a more general concept of an algebra A over S. If A has an infinite basis over S, both concepts can be generalized in another direction, which may be illustrated by rings of formal power series. At third, certain semirings A constructed from semirings S have become important tools e.g. in the theory of automata and formal languages, in particular again those in which also infinite sums occur. The purpose of this paper is to deal with all these different concepts in a unique way as special cases of generalized semialgebras over semirings. Since we have to describe all the material we want to combine, some parts of this paper have the character of a survey article.


Structure Constant Left Identity Formal Power Series Cardinal Number Associative Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Cohn, P.M., Some remarks on the invariant basis property, Topology 5 (1966), 215–228.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Eilenberg, S., Automata, Languages and Machines, Vol. A, Academic Press, 1974.Google Scholar
  3. [3]
    Everett, C.J., Vector Spaces over Rings, Bull. Amer. Math. Soc. 48 (1942), 312–316.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Griepentrog, R.D. and Weinert, H.J., Embedding semirings into semirings with identity (Proc. Conf. Szeged, 1976), Coll. Math. Soc. J. Bolyai 20 (1979), 225–245.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Higgs, D., Axiomatic infinite sums-an algebraic approach to integration theory, Contemp. Math. 2 (1980), 205–212.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Iizuka, K., On the Jacobson radical of a semiring, Tôhoku Math. J. (2) 11 (1959), 409–421.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Kuich, W. and Salomaa, A., Semirings, Automata, Languages, Springer-Verlag, 1986.Google Scholar
  8. [8]
    Lallement, G., Semigroups and Combinatiorial Applications, John Wiley & Sons, 1979.Google Scholar
  9. [9]
    Mahr, B., Semirings and transitive closure, TU Berlin, FB 20, Bericht-Nr. 82–5 (1982).Google Scholar
  10. [10]
    Mahr, B., Iteration and summability in semirings, Ann. Discrete Math. 19 (1984), 229–256.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Main, M.G. and Benson, D.B., Functional behavior of nondeterministic programs, Lecture Notes in Comp. Sci. 158 (1983), 290–301.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Manes, E.G. and Benson, D.B., The inverse semigroup of a sumordered semiring, Semigroup Forum 31 (1985), 129–152.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Pickert, G., Bemerkungen zum Algebrenbegriff, Math. Ann. 120 (1947–1949), 158–164.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Rédei, L., Algebra, Geest & Portig, 1959 (Engl. Ed. 1967).Google Scholar
  15. [15]
    Rote, G., A Systolic Array Algorithm for the Algebraic Path Problem, Computing 34 (1985), 191–219.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Salomaa, A. and Soittola, M., Automata-Theoretic Aspects of Formal Power Series, Springer-Verlag, 1978.Google Scholar
  17. [17]
    Steinfeld, O., Über die Struktursätze der Semiringe, Acta Math. Acad. Sci. Hungar. 10 (1959), 149–155.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Weinert, H.J., Über Halbringe und Halbkörper I, Acta Math. Acad. Sci. Hungar. 13 (1962), 365–378.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Weinert, H.J., On 0-simple semirings, semigroup semirings, and two kinds of division semirings, Semigroup Forum 28 (1984), 313–333.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Wongseelashote, A., Semirings and path spaces, Discrete Math. 26 (1979), 55–78.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Zassenhaus, H., Lehrbuch der Gruppentheorie, Teubner-Verlag, 1937 (Engl. Ed. Chelsea, 1949).Google Scholar

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© Springer-Verlag 1988

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  • Hanns Joachim Weinert

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