Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



A Stone-type representation theorem for algebras of relations of higher rank

Authors: H. Andréka and R. J. Thompson
Journal: Trans. Amer. Math. Soc. 309 (1988), 671-682
MSC: Primary 03G15; Secondary 03C95, 03G25
MathSciNet review: 961607
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [H] Andréka, A combinatorial proof for the famous Resek-Thompson theorem for cylindric algebras, Math. Inst. Hungar. Acad. Sci., Preprint, November 1986, 8 pp.
  • [W] Craig, Logic in algebraic form, North-Holland, 1974. MR 0411962 (54:91)
  • 1. -, Unification and abstraction in algebraic logic, Studies in Algebraic Logic, Vol. 9, Math. Assoc. Amer., Washington, D.C., 1974a, pp. 6-57. MR 0376345 (51:12521)
  • [M] Ferenczi, On the connection of cylindrical homomorphisms and point functions for $ {\operatorname{Crs} _\alpha }$ 's, Lectures in Universal Algebra (Proc. Szeged 1983), Colloq. Math. Soc. J. Bolyai, vol. 43, North-Holland, Amsterdam, 1985, pp. 123-141. MR 860260 (87m:03096)
  • [L] Henkin, Relativization with respect to formulas and its use in proofs of independence, Compositio Math. 20 (1968), 88-106. MR 0234812 (38:3126)
  • [L] Henkin and J. D. Monk, Cylindric algebras and related structures, Proc. Tarski Sympos., no. 25, Amer. Math. Soc., 1974, pp. 105-121. MR 0376346 (51:12522)
  • [HMTI] L. Henkin, J. D. Monk and A. Tarski, Cylindric algebras, Part I, North-Holland, 1971. MR 781929 (86m:03095a)
  • [HMTII] -, Cylindric algebras, Part II, North-Holland, 1985. MR 781930 (86m:03095b)
  • [HMTAN] L. Henkin, J. D. Monk, A. Tarski, H. Andréka and I. Németi, Cylindric set algebras, Lecture Notes in Math., vol. 883, Springer-Verlag, 1981. MR 639151 (84a:03078)
  • [L] Henkin and D. Resek, Relativization of cylindric algebras, Fund. Math. 82 (1975), 363-383. MR 0366659 (51:2906)
  • [B] Jónsson, Defining relations for full semigroups of finite transformations, Michigan Math. J. 9 (1962), 77-85. MR 0133390 (24:A3224)
  • [B] Jónsson and A. Tarski, Boolean algebras with operators. I, Amer. J. Math. 73 (1951), 891-939. MR 0044502 (13:426c)
  • [R] Maddux, Topics in relation algebras, Doctoral Dissertation, Berkeley, Calif., 1978.
  • 2. -, Some varieties containing relation algebras, Trans. Amer. Math. Soc. 272 (1982), 501-526. MR 662049 (84a:03079)
  • [I] 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.
  • 3. -, Cylindric-relativized set algebras have strong amalgamation, J. Symbolic Logic 50 (1985), 689-700. MR 805678 (87c:03146)
  • 4. -, Free algebras and decidability in algebraic logic, Doctoral Dissertation (B) for D.Sc (or Dr. Rer. Nat.), Hungar. Acad, of Sci., Budapest, 1986.
  • [C] Pinter, Cylindric algebras and algebras of substitutions, Trans. Amer. Math. Soc. 175 (1973), 167-179. MR 0317931 (47:6480)
  • [D] Resek, Some results on relativized cylindric algebras, Doctoral Dissertation, Berkeley, Calif., 1975.
  • [R] J. Thompson, Transformational structure of algebraic logics, Doctoral Dissertation, Berkeley, Calif., 1979.
  • 5. -, Defining relations for the semigroup of finite non-permutational transformations, Manuscript, Math. Inst. Hungar. Acad. Sci., Budapest, 1986, pp. 1-18.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03G15, 03C95, 03G25

Retrieve articles in all journals with MSC: 03G15, 03C95, 03G25

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society