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Minimal clans: A class of ordered partial semigroups including boolean rings and lattice-ordered groups

  • Klaus D. Schmidt
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1320)

Abstract

The present paper contains a rather comprehensive investigation of the properties of minimal clans — a new class of ordered partial semigroups which includes Boolean rings and lattice-ordered groups as special cases. It is shown that minimal clans preserve many properties of Boolean rings and lattice-ordered groups, that Boolean rings and lattice-ordered groups can be identified as minimal clans having, respectively, a minimal domain of addition or a maximal set of invertible elements, and that minimal clans in turn can be characterized in the classes of all symmetric clans, semiclans, and normal clans. Minimal clans are also compared with some other ordered algebraic structures with partial or complete addition that have been studied in the literature. An example of a minimal clan which is neither a Boolean ring nor a lattice-ordered group is provided by the collection of all fuzzy subsets of a given set.

Keywords

Fundamental Domain Fuzzy Subset Invertible Element Difference Property Jordan Decomposition 
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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References

  1. [1]
    Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et Anneaux Réticulés. Lecture Notes in Mathematics, vol. 608. Berlin-Heidelberg-New York: Springer 1977.zbMATHGoogle Scholar
  2. [2]
    Billhardt, B.: Zum Clan der normalen Teilbarkeitshalbgruppe. Dissertation. Kassel: Fachbereich Mathematik der Gesamthochschule Kassel 1981.zbMATHGoogle Scholar
  3. [3]
    Birkhoff, G.: Lattice-ordered groups. Ann. of Math. 43, 298–331 (1942).MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Birkhoff, G.: Lattice Theory. (Second) Revised Edition. Providence, Rhode Island: Amer. Math. Soc. 1948.zbMATHGoogle Scholar
  5. [5]
    Birkhoff, G.: Lattice Theory. Third (New) Edition. Providence, Rhode Island: Amer. Math. Soc. 1967.zbMATHGoogle Scholar
  6. [6]
    Bosbach, B.: Zur Theorie der Teilbarkeitschalbgruppen. Semigroup Forum 3, 1–30 (1971).MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Bosbach, B.: Schwache Teilbarkeitshalbgruppen. Semigroup Forum 12, 119–135 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Bosbach, B.: Concerning semiclans. Arch. Math. 37, 316–324 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Bosbach, B.: Lattice ordered binary systems. Mathematische Schriften Kassel, Preprint Nr. 4/84. Kassel: Fachbereich Mathematik der Gesamthochschule Kassel 1984.zbMATHGoogle Scholar
  10. [10]
    Brandt, H.: Über eine Verallgemeinerung des Gruppenbegriffes. Math. Ann. 96, 360–366 (1927).MathSciNetCrossRefGoogle Scholar
  11. [11]
    Brehmer, S.: Algebraic characterisation of measure and integral by the method of Caratheodory. In: Proc. Conf. Topology and Measure (Zinnowitz 1974), Part 1, pp. 23–53. Greifswald: Ernst-Moritz-Arndt-Universität 1978.Google Scholar
  12. [12]
    Clifford, A.H., Preston, G.B.: The Algebraic Theory of Semigroups, vol. I. Providence, Rhode Island: Amer. Math. Soc. 1961.CrossRefzbMATHGoogle Scholar
  13. [13]
    Conrad, P.: Generalized semigroup rings. J. Indian Math. Soc. (N.S.) 21, 73–95 (1957).MathSciNetzbMATHGoogle Scholar
  14. [14]
    Dinges, H.: Zur Algebra der Maßtheorie. Bull. Greek Math. Soc. 19, 25–97 (1978).MathSciNetzbMATHGoogle Scholar
  15. [15]
    Fuchs, L.: Teilweise geordnete algebraische Strukturen. Göttingen: Vandenhoeck & Ruprecht 1966.zbMATHGoogle Scholar
  16. [16]
    Ljapin, E.S.: Partielle Operationen in der Theorie der Halbgruppen. In: Semigroups. Lecture Notes in Mathematics, vol. 855, pp. 33–48. Berlin-Heidelberg-New York: Springer 1981.CrossRefGoogle Scholar
  17. [17]
    Nakano, T.: Rings and partly ordered systems. Math. Z. 99, 355–376 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Nickel, K.: Verbandstheoretische Grundlagen der Intervall-Mathematik. In: Interval Mathematics. Lecture Notes in Computer Science, vol. 29, pp. 251–262. Berlin-Heidelberg-New York: Springer 1975.CrossRefGoogle Scholar
  19. [19]
    Nickel, K.: Intervall-Mathematik. Z. Angew. Math. Mech. 58, T72–T85 (1978).MathSciNetzbMATHGoogle Scholar
  20. [20]
    Rama Rao, V.V.: On a common abstraction of Boolean rings and lattice ordered groups I. Monatsh. Math. 73, 411–421 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Riesz, F.: Sur quelques notions fondamentales dans la théorie générale des opérations linéaires. Ann. of Math. 41, 174–206 (1940).MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Schelp, R.H.: A partial semigroup approach to partially ordered sets. Proc. London Math. Soc. (3) 24, 46–58 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Schmidt, K.D.: A general Jordan decomposition. Arch. Math. 38, 556–564 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Schmidt, K.D.: A common abstraction of Boolean rings and lattice ordered groups. Comp. Math. 54, 51–62 (1985).MathSciNetzbMATHGoogle Scholar
  25. [25]
    Schmidt, K.D.: Embedding theorems for cones and applications to classes of convex sets occurring in interval mathematics. In: Interval Mathematics 1985. Lecture Notes in Computer Science, vol. 212, pp. 159–173. Berlin-Heidelberg-New York: Springer 1986.CrossRefGoogle Scholar
  26. [26]
    Stone, M.H.: Postulates for Boolean algebras and generalized Boolean algebras. Amer. J. Math. 57, 703–732 (1935).MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Stone, M.H.: The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40, 37–111 (1936).MathSciNetzbMATHGoogle Scholar
  28. [28]
    Swamy, K.L.N.: Dually residuated lattice ordered semigroups. Math. Ann. 159, 105–114 (1965).MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Wyler, O.: Clans. Comp. Math. 17, 172–189 (1966).MathSciNetzbMATHGoogle Scholar
  30. [30]
    Zadeh, L.A.: Fuzzy Sets. Inform. Control 8, 338–353 (1965).MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag 1988

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  • Klaus D. Schmidt

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