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algebra universalis

, Volume 30, Issue 2, pp 177–216 | Cite as

Power structures

  • Chris Brink
Article

Abstract

Power structures epitomise the attempt to lift some existing structure from elements of a set to subsets of that set. For example, any operationf over the elements of a setA can in a natural way be extended to apower operation f+ over subsets ofA, and hence there is for any algebraA a correspondingpower algebraP(A) (also called the “complex algebra” or “global” ofA), which is the power set endowed with the power operations. This idea goes back to Frobenius. A natural generalisation is to define for any (n + l)-ary relationR overA ann-ary operationR overP(A), and hence to form also the power algebra of any relational structure. Jónsson and Tarski first made systematic use of this construction in the study of Boolean algebras with operators. A further generalisation is to define for any relationR over a setA itspower relation R+ overP(A), and hence to form thepower structure of any relational structure. Power relations have been used, for example, in denotational semantics, in fixed-point theory, and in the study of verisimilitude. This paper offers an overview of known work and a pilot study of power structures in a universal-algebraic context.

Keywords

Pilot Study Power Relation Relational Structure Natural Generalisation Power Structure 
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]
    Alchourron, C. E., Gärdenfors, P. andMakinson, D.,On the logic of theory change: partial meet contraction and revision function, Journal of Symbolic Logic50 (1985), 510–530.Google Scholar
  2. [2]
    Almeida, J.,Power pseudovarities of semigroups I, II Semigroup Forum33 (1986), 357–373, 375–390.Google Scholar
  3. [3]
    Birkhoff, G.,Lattice Theory, AMS Colloquium Publications Vol. 25, Rhode Island, 1948.Google Scholar
  4. [4]
    Bleicher, M. N., Schneider, H. andWilson, R. L.,Permanence of identities on algebras, Algebra Universalis3 (1973), 72–93.Google Scholar
  5. [5]
    Bogdanovich, S.,A note on power semigroups, Math. Japon.28 (1983), 725–727.Google Scholar
  6. [6]
    Brink, C.,Boolean modules, J. Algebra71 (1981), 291–313.Google Scholar
  7. [7]
    Brink, C.,Power structures and logic, Quaestiones Math.9 (1986), 69–94.Google Scholar
  8. [8]
    Brink, C.,R -algebras and R -model structures as power constructs, Studia Logica48 (1989), 85–109.Google Scholar
  9. [9]
    Brink, C. andHeidema, J.,A verisimilar ordering of theories phrased in a propositional language, Brit. J. Philos. Sci.38 (1987), 533–549.Google Scholar
  10. [10]
    Bull, R. andSegerberg, K.,Basic modal logic, inHandbook of Philosophical Logic, Vol. II, Ed. D. Gabbay and F. Guenthner, Reidel, 1984, 1–88.Google Scholar
  11. [11]
    Comer, S. D.,Integral relation algebras via pseudogroups, Notices Amer. Math. Soc.23 (1976), A659.Google Scholar
  12. [12]
    Comer,A new foundation for the theory of relations, Notre Dame J. Formal Logic24 (1983), 181–187.Google Scholar
  13. [13]
    Dijkstra, E. W.,A Discipline of Programming, Prentice-Hall, 1976.Google Scholar
  14. [14]
    Fuhrmann, A.,Relevant Logics, Modal Logics and Theory Change, Ph.D. thesis, Automated Reasoning Project, Australian National University, 1988.Google Scholar
  15. [15]
    Gautam, N. D.,The validity of equations of complex algebras, Arch. Math. Logik Grundlag.3 (1957), 117–124.Google Scholar
  16. [16]
    Goldblatt, R.,An algebraic study of well-foundedness, Studia Logica44 (1985), 423–437.Google Scholar
  17. [17]
    Goldblatt, R.,Varieties of complex algebras, Annals of Pure and Applied Logic44 (1989), 173–242.Google Scholar
  18. [18]
    Grätzer, G.,Universal Algebra, 2nd ed., Springer-Verlag, 1979.Google Scholar
  19. [19]
    Grätzer, G. andLakser, H.,Identities for globals (complex algebras) of algebras, Colloq. Math.54 (1988), 19–29.Google Scholar
  20. [20]
    Grätzer, G. andWhitney, S.,Infinitary varieties of structures closed under the formation of complex structures, Colloq. Math.48 (1984), 1–5.Google Scholar
  21. [21]
    Gries, D.,The Science of Programming, Springer-Verlag, Berlin, 1983.Google Scholar
  22. [22]
    Grzegorczyk, A.,Some relational systems and the associated topological spaces, Fund. Math,60 (1967), 279–287.Google Scholar
  23. [23]
    Hall, M.,The Theory of Groups, MacMillan, New York, 1959.Google Scholar
  24. [24]
    Harel, D.,Dynamic logic, inHandbook of Philosophical Logic, Vol. II, Ed. D. Gabbay and F. Guenthner, Reidel, 1984, 497–604.Google Scholar
  25. [25]
    Ježek, J.,A note on complex groupoids, Colloq. Math. Soc. János Bolyai (1982), 419–420.Google Scholar
  26. [26]
    Jónsson, B.,Varieties of relation algebras, Algebra Universalis15 (1982), 273–298.Google Scholar
  27. [27]
    Jónsson, B. andTarski, A.,Boolean algebras with operators I, II, Amer. J. Math.73 (1951), 891–939;74 (1952), 127–167.Google Scholar
  28. [28]
    Kannai, Y. andPeleg, B.,A note on the extension of an order on a set to the power set, J. Econom. Theory32 (1984), 172–175.Google Scholar
  29. [29]
    Lemmon, E. J.,Algebraic semantics for modal logics, J. Symbolic Logic31 (1966), 46–65, 191–218.Google Scholar
  30. [30]
    MacDonald, I. D.,The Theory of Groups, Clarendon Press, Oxford, 1968.Google Scholar
  31. [31]
    Main, M. G.,A powerdomain primer, Bulletin of the EATCS33 (1987).Google Scholar
  32. [32]
    Margolis, S. W.,On M-varieties generated by power monoids, Semigroup Forum22 (1981), 339–353.Google Scholar
  33. [33]
    McCarthy, D. J. andHayes, D. L.,Subgroups of the power semigroup of a group, J. Combin. Theory Ser. A14 (1973), 173–186.Google Scholar
  34. [34]
    McKenzie, R.,Representation of integral relation algebras, Michigan Math. J.17 (1970), 279–287.Google Scholar
  35. [35]
    Plotkin, G. D.,Dijkstra's predicate transformers and Smyth's power-domains, Proceedings of the 1979 Copenhagen Winter School in Abstract Software Specifications, Ed. D. Bjorner, Lecture Notes in Compute Science86 (1980), 525–553.Google Scholar
  36. [36]
    Putcha, M. S.,On the maximal semilattice decomposition of the power semigroup of a semigroup, Semigroup Forum15 (1978), 263–267.Google Scholar
  37. [37]
    Ratschek, H.,Universal inclusion structures, Colloq. Math. Soc. János Bolyai (1982), 625–633.Google Scholar
  38. [38]
    Scott, D. S.,Domains for denotational semantics, ICALP 9, Ed. M. Nielsen and E. M. Schmidt, Lecture Notes in Computer Science140 (1982), 577–613.Google Scholar
  39. [39]
    Shafaat, A.,On varieties closed under the construction of power algebras, Bull. Austral. Math. Soc.11 (1974), 213–218.Google Scholar
  40. [40]
    Shafaat, A.,Remarks on special homomorphic relations, Period. Math. Hungar.6 (1975), 255–265.Google Scholar
  41. [41]
    Shafaat, A.,Constructions preserving the associative and commutative laws, J. Austral. Math. Soc.21 (1976), 112–117.Google Scholar
  42. [42]
    Shafaat, A.,Homomorphisms, homomorphic relations and power algebras, Per. Math. Hungar.11 (1980), 89–94.Google Scholar
  43. [43]
    Smithson, R. E.,Fixed points of order preserving multifunctions, Proc. Amer. Math. Soc.28 (1971), 304–310.Google Scholar
  44. [44]
    Smyth, M. B.,Power domains and predicate transformers: a topological view, ICALP 10, Ed. J. Diaz, Lecture Notes in Computer Science154 (1983), 662–675.Google Scholar
  45. [45]
    Straubing, H.,Recognizable sets and power sets of finite semigroups, Semigroup Forum18 (1979), 331–340.Google Scholar
  46. [46]
    Szendrei, A.,The operation ISKP on classes of algebras, Algebra Universalis6 (1976), 349–353.Google Scholar
  47. [47]
    Tamura, T.,Power semigroups of rectangular groups, Math. Japon.29 (1984), 671–678.Google Scholar
  48. [48]
    Tamura, T. andShafer, J.,Power semigroups, Math. Japon.12 (1967), 25–32.Google Scholar
  49. [49]
    Trnkova, V.,On a representation of commutative semigroups, Semigroup Forum10 (1975), 203–214.Google Scholar
  50. [50]
    Van Benthem, J.,Correspondence theory, inHandbook of Philosophical Logic, Vol. II, Ed. P. Gabbay and F. Guenthner, Reidel, 1984, 167–247.Google Scholar
  51. [51]
    Walker, J.,Isotone relations and the fixed-point property for posets, Discrete Math.48 (1984), 275–288.Google Scholar
  52. [52]
    Whitney, S.,Théories linéaries, Ph.D. thesis, Université Laval, Québec, 1977.Google Scholar
  53. [53]
    Winskel, G.,On powerdomains and modality, Theoret. Comp. Sci.36 (1985), 127–137.Google Scholar

Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • Chris Brink
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of Cape TownRondeboschSouth Africa
  2. 2.Automated Reasoning Project Centre for Information Science ResearchAustralian National UniversityCanberraAustralia

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