Abstract
Given a simple atomic relation algebra \({\mathcal{A}}\) and a finite n ≥ 3, we construct effectively an atomic n-dimensional polyadic equality-type algebra \({\mathcal{P}}\) such that for any subsignature L of the signature of \({\mathcal{P}}\) that contains the boolean operations and cylindrifications, the L-reduct of \({\mathcal{P}}\) is completely representable if and only if \({\mathcal{A}}\) is completely representable. If \({\mathcal{A}}\) is finite then so is \({\mathcal{P}}\) .
It follows that there is no algorithm to determine whether a finite n-dimensional cylindric algebra, diagonal-free cylindric algebra, polyadic algebra, or polyadic equality algebra is representable (for diagonal-free algebras this was known). We also obtain a new proof that the classes of completely representable n-dimensional algebras of these types are non-elementary, a result that remains true for infinite dimensions if the diagonals are present, and also for infinite-dimensional diagonal-free cylindric algebras.
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References
Chang C.C., Keisler H.J.: Model Theory, 3rd edn. North-Holland, Amsterdam (1990)
Goldblatt R.: Varieties of complex algebras. Ann. Pure Appl. Logic 44, 173–242 (1989)
Goldblatt R.: Elementary generation and canonicity for varieties of boolean algebras with operators. Algebra Universalis 34, 551–607 (1995)
Henkin L., Monk J.D., Tarski A. (1971) Cylindric Algebras Part I. North-Holland
Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras Part II. North-Holland (1985)
Hirsch R., Hodkinson I.: Complete representations in algebraic logic. J. Symbolic Logic 62, 816–847 (1997)
Hirsch R., Hodkinson I.: Representability is not decidable for finite relation algebras. Trans. Amer. Math. Soc. 353, 1403–1425 (2001)
Hirsch R., Hodkinson I.: Relation Algebras by Games. Studies in Logic and the Foundations of Mathematics, vol. 147. Amsterdam, North-Holland (2002)
Hirsch R., Hodkinson I.: Strongly representable atom structures of cylindric algebras. J. Symbolic Logic 74, 811–828 (2009)
Hirsch R., Hodkinson I., Kurucz A.: On modal logics between K × K × K and S5 × S5 × S5. J. Symbolic Logic 67, 221–234 (2002)
Hodkinson, I., Wolter, F., Zakharyaschev, M.: Decidable and undecidable fragments of first-order branching temporal logics. In: Proc. 17th IEEE Symposium on Logic in Computer Science (LICS), pp. 393–402. IEEE Inc. (2002)
Johnson J.S.: Nonfinitizability of classes of representable polyadic algebras. J. Symbolic Logic 34, 344–352 (1969)
Khaled M., Sayed Ahmed T.: On complete representations of algebras of logic. Logic J. IGPL 17, 267–272 (2009)
Kurucz, A.: On the complexity of modal axiomatisations over many-dimensional structures. In: Beklemishev, L., Goranko, V., Shehtman, V. (eds.) Advances in Modal Logic, vol. 8, pp. 256–270. College Publications (2010)
Lyndon R.: The representation of relational algebras. Ann. Math. 51, 707–729 (1950)
Maddux, R.D. (1978) Topics in Relation Algebra. PhD thesis, University of California, Berkeley
Maddux R.D.: Non-finite axiomatizability results for cylindric and relation algebras. J. Symbolic Logic 54, 951–974 (1989)
Maddux, R.D.: Introductory course on relation algebras, finite-dimensional cylindric algebras, and their interconnections. In: Andréka, H., Monk, J.D., Németi, I. (eds.) Algebraic Logic. Colloq. Math. Soc. J. Bolyai, vol. 54, pp. 361–392. North-Holland, Amsterdam (1991)
Maddux R.D.: Relation Algebras. Studies in Logic and the Foundations of Mathematics, vol. 150. Elsevier, Amsterdam (2006)
Monk, J.D.: Studies in Cylindric Algebra. PhD thesis, University of California, Berkeley (1961)
Monk J.D.: On representable relation algebras. Michigan Math. J. 11, 207–210 (1964)
Monk J.D.: Nonfinitizability of classes of representable cylindric algebras. J. Symbolic Logic 34, 331–343 (1969)
Németi, I.: Free Algebras and Decidability in Algebraic Logic. Doctoral Dissertation with the Academy, Budapest (1986)
Pinter C.: Cylindric algebras and algebras of substitutions. Trans. Amer. Math. Soc. 175, 167–179 (1973)
Sági G.: A note on algebras of substitutions. Studia Logica 72, 265–284 (2002)
Tarski A.: On the calculus of relations. J. Symbolic Logic 6, 73–89 (1941)
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Presented by J. Raftery.
The research was partially supported by UK EPSRC grant GR/S19905/01.
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Hodkinson, I. A construction of cylindric and polyadic algebras from atomic relation algebras. Algebra Univers. 68, 257–285 (2012). https://doi.org/10.1007/s00012-012-0202-3
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DOI: https://doi.org/10.1007/s00012-012-0202-3