Notre Dame Journal of Formal Logic

Neat Embeddings, Omitting Types, and Interpolation: An Overview

Tarek Sayed Ahmed

Abstract

We survey various results on the relationship among neat embeddings (a notion special to cylindric algebras), complete representations, omitting types, and amalgamation. A hitherto unpublished application of algebraic logic to omitting types of first-order logic is given.

Article information

Source
Notre Dame J. Formal Logic Volume 44, Number 3 (2003), 157-173.

Dates
First available in Project Euclid: 28 July 2004

http://projecteuclid.org/euclid.ndjfl/1091030854

Digital Object Identifier
doi:10.1305/ndjfl/1091030854

Mathematical Reviews number (MathSciNet)
MR2130788

Zentralblatt MATH identifier
1071.03041

Citation

Sayed Ahmed, Tarek. Neat Embeddings, Omitting Types, and Interpolation: An Overview. Notre Dame J. Formal Logic 44 (2003), no. 3, 157--173. doi:10.1305/ndjfl/1091030854. http://projecteuclid.org/euclid.ndjfl/1091030854.

References

• [1] Andréka, H., "Atomic representable relation and cylindric algebras with nonrepresentable completions", unpublished manuscript.
• [2] Andréka, H., J. Comer, I. Németi, and T. Sayed Ahmed, "Epimorphisms in cylindric algebras", Preprint.
• [3] Andréka, H., T. Gergely, and I. Németi, "On universal algebraic constructions of logics", Studia Logica, vol. 36 (1977), pp. 9–47.
• [4] Andréka, H., J. D. Monk, and I. Németi, editors, Algebraic Logic, vol. 54 of Colloquia Mathematica Societatis János Bolyai, North-Holland Publishing Co., Amsterdam, 1991. Papers from the colloquium held in Budapest, 1988.
• [5] Andréka, H., and I. Németi, "On systems of varieties definable by schemes of equations", Algebra Universalis, vol. 11 (1980), pp. 105–16.
• [6] Andréka, H., and T. Sayed Ahmed, "Omitting types in logics with finitely many variables", Bulletin of Symbolic Logic, vol. 5 (1999), p. 88.
• [7] Andréka, H., "Complexity of equations valid in algebras of relations I. Strong non-finitizability", Annals of Pure and Applied Logic, vol. 89 (1997), pp. 149–209.
• [8] Biró, B., "Non-finite-axiomatizability results in algebraic logic", The Journal of Symbolic Logic, vol. 57 (1992), pp. 832–43.
• [9] Blok, W. J., and D. Pigozzi, "Algebraizable logics", Memoirs of the American Mathematical Society, vol. 77 (1989).
• [10] Casanovas, E., and R. Farré, "Omitting types in incomplete theories", The Journal of Symbolic Logic, vol. 61 (1996), pp. 236–45.
• [11] Chang, C. C., and H. J. Keisler, Model Theory, 3d edition, vol. 73 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990.
• [12] Goldblatt, R., "Varieties of complex algebras", Annals of Pure and Applied Logic, vol. 44 (1989), pp. 173–242.
• [13] Goldblatt, R., "Persistence and atomic generation for varieties of Boolean algebras with operators", Studia Logica, vol. 68 (2001), pp. 155–71.
• [14] Henkin, L., J. D. Monk, and A. Tarski, Cylindric Algebras. Part I, vol. 64 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1971.
• [15] Henkin, L., J. D. Monk, and A. Tarski, Cylindric Algebras. Part II, vol. 115 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1985.
• [16] Hirsch, R., I. Hodkinson, and A. Kurucz, "On modal logics between $\mathrm{K}\times\mathrm{K}\times \mathrm{K}$ and S5 $\times$ S5 $\times$ S5", The Journal of Symbolic Logic, vol. 67 (2002), pp. 221–34.
• [17] Hirsch, R., and T. Sayed Ahmed, "Neat embeddability is not sufficient for complete representations", 2001.
• [18] Hirsch, R., and I. Hodkinson, "Complete representations in algebraic logic", The Journal of Symbolic Logic, vol. 62 (1997), pp. 816–47.
• [19] Hirsch, R., and I. Hodkinson, "Step by step–-building representations in algebraic logic", The Journal of Symbolic Logic, vol. 62 (1997), pp. 225–79.
• [20] Hirsch, R., and I. Hodkinson, "Relation algebras from cylindric algebras. II", Annals of Pure and Applied Logic, vol. 112 (2001), pp. 267–97.
• [21] Hirsch, R., and I. Hodkinson, Relation Algebras by Games, vol. 147 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 2002.
• [22] Hirsch, R., I. Hodkinson, and R. D. Maddux, "Provability with finitely many variables", The Bulletin of Symbolic Logic, vol. 8 (2002), pp. 348–79.
• [23] Hirsch, R., I. Hodkinson, and R. D. Maddux, "Relation algebra reducts of cylindric algebras and an application to proof theory", The Journal of Symbolic Logic, vol. 67 (2002), pp. 197–213.
• [24] Hodges, W., Model Theory, vol. 42 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993.
• [25] Hodkinson, I., "Atom structures of cylindric algebras and relation algebras", Annals of Pure and Applied Logic, vol. 89 (1997), pp. 117–48.
• [26] Lyndon, R. C., "Relation algebras and projective geometries", Michigan Mathematical Journal, vol. 8 (1961), pp. 21–28.
• [27] Madarász, J., and T. Sayed Ahmed, "Amalgamation, interpolation and epimorphisms, solutions to all problems of Pigozzi's paper, and some more", Preprint of the Mathematical Institute of the Hungarian Academy of Sciences.
• [28] Maddux, R. D., "Nonfinite axiomatizability results for cylindric and relation", The Journal of Symbolic Logic, vol. 54 (1989), pp. 951–74.
• [29] Maksimova, L., "Amalgamation and interpolation in normal modal logic", Studia Logica, vol. 50 (1991), pp. 457–71.
• [30] Monk, J. D., "Nonfinitizability of classes of representable cylindric algebras", The Journal of Symbolic Logic, vol. 34 (1969), pp. 331–43.
• [31] Monk, J. D., "An introduction to cylindric set algebras", Logic Journal of the IGPL, vol. 8 (2000), pp. 451–96.
• [32] Németi, I., "Cylindric-relativised set algebras have strong amalgamation", The Journal of Symbolic Logic, vol. 50 (1985), pp. 689–700.
• [33] Németi, I., "The class of neat-reducts of cylindric algebras is not a variety but is closed with respect to ${\rm HP}$", Notre Dame Journal of Formal Logic, vol. 24 (1983), pp. 399–409.
• [34] Newelski, L., "Omitting types and the real line", The Journal of Symbolic Logic, vol. 52 (1987), pp. 1020–26.
• [35] Pigozzi, D., "Amalgamation, congruence-extension, and interpolation properties in algebras", Algebra Universalis, vol. 1 (1971/72), pp. 269–349.
• [36] Sain, I., "On the search for a finitizable algebraization of first order logic", Logic Journal of the IGPL, vol. 8 (2000), pp. 497–591.
• [37] Sain, I., and V. Gyuris, "Finite schematizable algebraic logic", Logic Journal of the IGPL, vol. 5 (1997), pp. 699–751 (electronic).
• [38] Sayed Ahmed, T., On Neat Reducts of Algebras of Logic, Ph.D. thesis, Cairo University, 2002.
• [39] Sayed Ahmed, T., "A categorial approach to amalgamation theorems", unpublished manuscript.
• [40] Sayed Ahmed, T., "The class of neat reducts is a psuedo-elementary class that is not closed under ultraroots", unpublished manuscript.
• [41] Sayed Ahmed, T., "On amalgamation of algebras of logic, a functorial formulation", unpublished manuscriopt.
• [42] Sayed Ahmed, T., "On amalgamation of reducts of polyadic algebras", forthcoming in Algebra Universalis, http://www.math-inst.hu/pub/algebraic-logic.
• [43] Sayed Ahmed, T., "The class of neat reducts is not elementary", The Bulletin of Symbolic Logic, vol. 5 (1999), pp. 407–408.
• [44] Sayed Ahmed, T., On a stronger version of the finitization problem, Alfred Rényi Institute of Mathematics, Budapest, 1999. http://www.renyi.hu/\~ rabbit/.
• [45] Sayed Ahmed, T., "The class of neat reducts is not elementary", Logic Journal of the IGPL, vol. 9 (2001), pp. 593–628. http://www.math-inst.hu/pub/algebraic-logic.
• [46] Sayed Ahmed, T., "The class of 2-dimensional neat reducts is not elementary", Fundamenta Mathematicae, vol. 172 (2002), pp. 61–81.
• [47] Sayed Ahmed, T., "Martin's axiom, omitting types, and complete representations in algebraic logic", Studia Logica, vol. 72 (2002), pp. 285–309.
• [48] Sayed Ahmed, T., "A model-theoretic solution to a problem of Tarski", Mathematical Logic Quarterly, vol. 48 (2002), pp. 343–55. http://www.interscience.wiley.com.
• [49] Sayed Ahmed, T., "A confirmation of a conjecture of Tarski", Bulletin of the Section of Logic, University of \Lódź, vol. 32 (2003), pp. 103–105.
• [50] Sayed Ahmed, T., "Omitting types for algebraizable extensions of first order logic", unpublished manuscript.
• [51] Sayed Ahmed, T., "RCA$_{\omega}$ is not Sahlqvist, and is not closed under minimal completions", unpublished manuscript.
• [52] Sayed Ahmed, T., “Which reducts of polyadic algebras are atom-canonical?” unpublished manuscript.
• [53] Sayed Ahmed, T., and I. Németi, "On neat reducts of algebras of logic", Studia Logica, vol. 68 (2001), pp. 229–62.
• [54] Venema, Y., "Cylindric modal logic", The Journal of Symbolic Logic, vol. 60 (1995), pp. 591–623.
• [55] Venema, Y., "Rectangular games", The Journal of Symbolic Logic, vol. 63 (1998), pp. 1549–64.
• [56] Weaver, G., and J. Welaish, "Back and forth constructions in modal logic: An interpolation theorem for a family of modal logics", The Journal of Symbolic Logic, vol. 51 (1986), pp. 969–80.