Stability and Categoricity of Lattices

Smith, Kenneth W.

Canadian journal of mathematics, Tome 33 (1981) no. 6, pp. 1380-1419

Résumé

This paper is a contribution to applied stability theory. Our purpose is to investigate the complexity of lattices by determining the stability of their first order theories.Stability measures the complexity of a theory T by counting the number of different “kinds” of elements in models of T. The notion of ω-stability was introduced by Morley [26] in 1965 and generalized by Shelah [31] in 1969. Shelah classified all first order theories according to their stability properties.Stability and -categoricity are closely related (see [26] and [1]). In fact, the notions of stable, superstable and ω-stable can be regarded as successive approximations of -categorical. -categoricity is a very strong property while stability, superstability and ω-stability facilitate the classification of more “complex” theories.

Détail

DOI : 10.4153/CJM-1981-107-3

Bibliographie

[1] 1. Baldwin, J. T. and Lachlan, A. H., On strongly minimal sets, J. Symbolic Logic 36 (1971), 79–96. Google Scholar

[2] 2. Barwise, J., Back and forth through infinitary logic, in: Studies in model theory, Math. Assoc. Amer., Buffalo (1973), 5–34. Google Scholar

[3] 3. Barwise, J., Handbook of mathematical logic, (North-Holland, Amsterdam, 1977). Google Scholar

[4] 4. Chang, C. C. and Keisler, H. J., Model theory, (North-Holland, Amsterdam, 1973). Google Scholar

[5] 5. Cherlin, G., Model theoretic algebra—selected topics, Lecture Notes in Mathematics 521 (Springer-Verlag, Berlin, 1976). Google Scholar

[6] 6. Dickmann, M. A., Large infinitary languages, (North-Holland, Amsterdam, 1975). Google Scholar

[7] 7. Dushnik, B. and Miller, E. W., Partially ordered sets. Amer. J. Math. 63 (1941), 600–610. Google Scholar

[8] 8. Eklof, P. C. and Fisher, E. R., The elementary theory of abelian groups, Annals Math. Logic 4 (1972), 115–171. Google Scholar

[9] 9. Ehrenfeucht, A., An application of games to the completeness problem for formalized theories, Fund. Math. 49 (1961), 129–141. Google Scholar

[10] 10. Feferman, S. and Vaught, R. L., The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57–103. Google Scholar

[11] 11. Fennemore, C. F., All varieties of bands I and II, Math. Nach. 48 (1971), 237–262. Google Scholar

[12] 12. Fraïssé, R., Sur quelques classifications des systèmes de relations, Publ. Sci. Univ. Alger. Sér. A, 1 (1954), 35–182. Google Scholar

[13] 13. Gerhard, J. A., The lattice of equational classes of idempotent semigroups, J. Algebra 15 (1970), 195–224. Google Scholar

[14] 14. Grätzer, G., Universal algebra, (D. Van Nostrand, Princeton, 1968). Google Scholar

[15] 15. Hanf, W., Model-theoretic methods in the study of elementary logic, in: The theory of models, Proceedings of the 1963 International Symposium at Berkley (North-Holland, Amsterdam, 1965), 132–145. Google Scholar

[16] 16. Harnik, V., On the existence of saturated models of stable theories, Proc. Amer. Math. Soc. 52 (1975), 361–367. Google Scholar

[17] 17. Howie, J. M., An introduction to semigroup theory, (Academic Press, London, 1976). Google Scholar

[18] 18. Keisler, H. J., Fundamentals of model theory, in: Handbook of mathematical logic (North-Holland, Amsterdam, 1977), 47–103. Google Scholar

[19] 19. Kelley, D. and Rival, I., Planar lattices, Can. J. Math. 27 (1975), 636–665. Google Scholar

[20] 20. Kunen, K., Combinatorics, in: Handbook of mathematical logic (North-Holland, Amsterdam, 1977), 371–402. Google Scholar

[21] 21. Lachlan, A. H., On the number of countable models of a countable superstable theory, in: Logic, methodology and philosophy of science IV (North-Holland, Amsterdam, 1973), 45–56. Google Scholar

[22] 22. Lachlan, A. H., Two conjectures regarding the stability of ω-categorical theories, Fund. Math. 81 (1974), 133–145. Google Scholar

[23] 23. Macintyre, A., On ω-categorical theories of abelian groups, Fund. Math. 70 (1971), 253–270. Google Scholar

[24] 24. Mekler, A. and Smith, K. W., Superstable graphs, Not. Amer. Math. Soc. (1979), 79T–E15. Google Scholar

[25] 25. Monk, J. D., Mathematical logic, (Springer-Verlag, New York, 1976). Google Scholar

[26] 26. Morley, M., Categoricity in power. Trans. Amer. Math. Soc. 114 (1965), 514–538. Google Scholar

[27] 27. Olin, P., Logical properties of V-free products of bands, preprint. Google Scholar

[28] 28. Olin, P. and Smith, K. W., Stability and the underlying semilattice of a band, preprint. Google Scholar

[29] 29. Petrich, M., A construction and a classification of bands, Math. Nach. 48 (1971), 263–274. Google Scholar

[30] 30. Podewski, K. and Ziegler, M., Stable graphs, Fund. Math. 100 (1978), 101–107. Google Scholar

[31] 31. Shelah, S., Stable theories, Israel J. Math. 7 (1969), 187–202. Google Scholar

[32] 32. Shelah, S., Finite diagram stable in power, Annals Math. Logic 2 (1970), 69–118. Google Scholar

[33] 33. Shelah, S., Stability, the f.c.p. and superslability, Annals Math. Logic 3 (1971), 271–362. Google Scholar

[34] 34. Smith, K. W., Stability and categoricity of lattices of height four, Not. Amer. Math. Soc. (1978), 78T–E49. Google Scholar

[35] 35. Smith, K. W., Stability of lattices of finite height, Not. Amer. Math. Soc. (1979), 79T–E9. Google Scholar

[36] 36. Smith, K. W., Stability of lattices, Ph.D. Thesis, University of Toronto (1979). Google Scholar

[37] 37. Wierzejewski, J., Remarks on stability and saturated models, Coll. Math. 34 (1976), 165–169. Google Scholar