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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A Stone-type representation theorem for algebras of relations of higher rank
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by H. Andréka and R. J. Thompson PDF
Trans. Amer. Math. Soc. 309 (1988), 671-682 Request permission

Abstract:

The Stone representation theorem for Boolean algebras gives us a finite set of equations axiomatizing the class of Boolean set algebras. Boolean set algebras can be considered to be algebras of unary relations. As a contrast here we investigate algebras of $n$-ary relations (originating with Tarski). The new algebras have more operations since there are more natural set theoretic operations on $n$-ary relations than on unary ones. E.g. the identity relation appears as a new constant. The Resek-Thompson theorem we prove here gives a finite set of equations axiomatizing the class of algebras of $n$-ary relations (for every ordinal $n$).
References
    Andréka, A combinatorial proof for the famous Resek-Thompson theorem for cylindric algebras, Math. Inst. Hungar. Acad. Sci., Preprint, November 1986, 8 pp.
  • William Craig, Logic in algebraic form, Studies in Logic and the Foundations of Mathematics, Vol. 72, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. Three languages and theories. MR 0411962
  • William Craig, Unification and abstraction in algebraic logic, Studies in algebraic logic, Studies in Math., Vol. 9, Math. Assoc. Amer., Washington, D.C., 1974, pp. 6–57. MR 0376345
  • M. Ferenczi, On the connection of cylindrical homomorphisms and point functions for $\textrm {Crs}_\alpha$’s, Lectures in universal algebra (Szeged, 1983) Colloq. Math. Soc. János Bolyai, vol. 43, North-Holland, Amsterdam, 1986, pp. 123–141. MR 860260
  • Leon Henkin, Relativization with respect to formulas and its use in proofs of independence, Compositio Math. 20 (1968), 88–106 (1968). MR 234812
  • Leon Henkin and J. Donald Monk, Cylindric algebras and related structures, Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1974, pp. 105–121. MR 0376346
  • Leon Henkin, J. Donald Monk, and Alfred Tarski, Cylindric algebras. Part I, Studies in Logic and the Foundations of Mathematics, vol. 64, North-Holland Publishing Co., Amsterdam, 1985. With an introductory chapter: General theory of algebras; Reprint of the 1971 original. MR 781929
  • Leon Henkin, J. Donald Monk, and Alfred Tarski, Cylindric algebras. Part I, Studies in Logic and the Foundations of Mathematics, vol. 64, North-Holland Publishing Co., Amsterdam, 1985. With an introductory chapter: General theory of algebras; Reprint of the 1971 original. MR 781929
  • Leon Henkin, J. Donald Monk, Alfred Tarski, Hajnal Andréka, and István Németi, Cylindric set algebras, Lecture Notes in Mathematics, vol. 883, Springer-Verlag, Berlin-New York, 1981. MR 639151, DOI 10.1007/BFb0095612
  • Leon Henkin and Diane Resek, Relativization of cylindric algebras, Fund. Math. 82 (1974/75), 363–383. MR 366659, DOI 10.4064/fm-82-4-363-383
  • Bjarni Jónsson, Defining relations for full semigroups of finite transformations, Michigan Math. J. 9 (1962), 77–85. MR 133390
  • Bjarni Jónsson and Alfred Tarski, Boolean algebras with operators. I, Amer. J. Math. 73 (1951), 891–939. MR 44502, DOI 10.2307/2372123
    • Maddux, Topics in relation algebras, Doctoral Dissertation, Berkeley, Calif., 1978.
  • Roger Maddux, Some varieties containing relation algebras, Trans. Amer. Math. Soc. 272 (1982), no. 2, 501–526. MR 662049, DOI 10.1090/S0002-9947-1982-0662049-7
    • Németi, Connections between cylindric algebras and initial algebra semantics of $CF$ languages, Mathematical Logic in Computer Science (Proc. Colloq. Salgótarján 1978), Colloq. Math. Soc. J. Bolyai, vol. 26, North-Holland, 1981, pp. 561-605.
  • I. Németi, Cylindric-relativised set algebras have strong amalgamation, J. Symbolic Logic 50 (1985), no. 3, 689–700. MR 805678, DOI 10.2307/2274323
    • —, Free algebras and decidability in algebraic logic, Doctoral Dissertation (B) for D.Sc (or Dr. Rer. Nat.), Hungar. Acad, of Sci., Budapest, 1986.
  • Charles Pinter, Cylindric algebras and algebras of substitutions, Trans. Amer. Math. Soc. 175 (1973), 167–179. MR 317931, DOI 10.1090/S0002-9947-1973-0317931-1
    • Resek, Some results on relativized cylindric algebras, Doctoral Dissertation, Berkeley, Calif., 1975. J. Thompson, Transformational structure of algebraic logics, Doctoral Dissertation, Berkeley, Calif., 1979. —, Defining relations for the semigroup of finite non-permutational transformations, Manuscript, Math. Inst. Hungar. Acad. Sci., Budapest, 1986, pp. 1-18.
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 309 (1988), 671-682
  • MSC: Primary 03G15; Secondary 03C95, 03G25
  • DOI: https://doi.org/10.1090/S0002-9947-1988-0961607-5
  • MathSciNet review: 961607