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SPACES OF TYPES IN POSITIVE MODEL THEORY

Published online by Cambridge University Press:  24 January 2019

LEVON HAYKAZYAN
LEVON HAYKAZYAN*
Affiliation:
DEPARTMENT OF PURE MATHEMATICS UNIVERSITY OF WATERLOO 200 UNIVERSTITY AVENUE WEST WATERLOO, ONTARIO N2L 3G1, CANADAE-mail: lhaykazyan@uwaterloo.ca
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Abstract

We introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first-order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory.

Keywords

03C0703C1503C50positive model theoryStone dualityomitting typesatomic models

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 2 , June 2019 , pp. 833 - 848
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

REFERENCES

Belkasmi, M., Positive model theory and amalgamation. Notre Dame Journal of Formal Logic, vol. 55 (2014), no. 2, pp. 205–230.10.1215/00294527-2420648CrossRefGoogle Scholar
Ben-Yaacov, I., Positive model theory and compact abstract theories. Journal of Mathematical Logic, vol. 3 (2003), no. 1, pp. 85–118.10.1142/S0219061303000212CrossRefGoogle Scholar
Ben-Yaacov, I. and Poizat, B., Fondements de la logique positive, this Journal, vol. 72 (2007), no. 4, pp. 1141–1162.Google Scholar
Hochster, M., Prime ideal structure in commutative rings. Transactions of the American Mathematical Society, vol. 142 (1969), pp. 43–60.10.1090/S0002-9947-1969-0251026-XCrossRefGoogle Scholar
Hodges, W., Building Models by Games, Cambridge University Press, Cambridge, 1985.Google Scholar
Hofmann, K. H., A note on Baire spaces and continuous lattices. Bulletin of the Australian Mathematical Society, vol. 21 (1980), pp. 265–279.10.1017/S0004972700006080CrossRefGoogle Scholar
Hrushovski, E., Simplicity and the Lascar group, unpublished preprint, 1998.Google Scholar
Isbell, J. R., Function spaces and adjoins. Mathematica Scandinavica, vol. 36 (1975), pp. 317–339.10.7146/math.scand.a-11581CrossRefGoogle Scholar
Lascar, D., Stability in model theory, Longman Scientific & Technical, 1987, Translated from the French by J. E. Wallington.Google Scholar
Mac Lane, S. and Moerdijk, I., Sheaves in Geometry and Logic, Springer Verlag, New York, 1992.Google Scholar
Macintyre, A., Model theory: Geometrical and set-theoretic aspects and prospects. The Bulletin of Symbolic Logic, vol. 9 (2003), no. 2, pp. 197–212.10.2178/bsl/1052669289CrossRefGoogle Scholar
Morley, M., Applications of topology to ${L_{{\omega _1},\omega }}$., Proceedings of the Tarski Symposium (Henkin, L., Addison, J., Chang, C. C., Craig, W., Scott, D., and Vaught, R., editors), American Mathematical Society, Providence, RI, 1974, pp. 233–240.10.1090/pspum/025/0373879CrossRefGoogle Scholar
Pillay, A., Forking in the category of existentially closed structures, Connections Between Model Theory and Algebraic and Analytic Geometry (Macintyre, A., editor), Dipartimento di Matematica della Seconda Università di Napoli, Napoli, 2000, pp. 23–42.Google Scholar
Poizat, B., Quelques effets pervers de la positivité. Annals of Pure and Applied Logic, vol. 161 (2010), no. 6, pp. 812–816.10.1016/j.apal.2009.06.012CrossRefGoogle Scholar
Poizat, B. and Yeshkeyev, A., Positive Jonsson theories. Logica Universalis, vol. 12 (2018), no. 1–2, pp. 101–127.10.1007/s11787-018-0185-8CrossRefGoogle Scholar
Shelah, S., The lazy model-theoretician’s guide to stability. Logique et Analyse, vol. 18 (1975), no. 71–72, pp. 241–308.Google Scholar
Simmons, H., Large and small existentially closed structures, this Journal, vol. 41 (1976), no. 2, pp. 379–390.Google Scholar
Stone, M. H., Topological representations of distributive lattices and Brouwerian logics. Časopis pro pěstování matematiky a fysiky, vol. 67 (1938), no. 1, pp. 1–25.Google Scholar