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)

 
 

 

Vaught's conjecture for varieties


Authors: Bradd Hart, Sergei Starchenko and Matthew Valeriote
Journal: Trans. Amer. Math. Soc. 342 (1994), 173-196
MSC: Primary 03C45; Secondary 03C05, 03C60, 08B99
DOI: https://doi.org/10.1090/S0002-9947-1994-1191612-0
MathSciNet review: 1191612
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if $ \mathcal{V}$ is a superstable variety or one with few countable models then $ \mathcal{V}$ is the varietal product of an affine variety and a combinatorial variety. Vaught's conjecture for varieties is an immediate consequence.


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

  • [1] J. Baldwin and A. Lachlan, On universal Horn classes categorical in some infinite power, Algebra Universalis 3 (1973), 98-111. MR 0351785 (50:4273)
  • [2] J. Baldwin and R. McKenzie, Counting models in universal Horn classes, Algebra Universalis 15 (1982), 359-384. MR 689770 (84m:03042)
  • [3] W. Baur, $ {\aleph _0}$-categorical modules, J. Symbolic Logic 40 (1975), 213-220. MR 0369047 (51:5283)
  • [4] Steve Buechler and Saharon Shelah, On the existence of regular types, Ann. Pure Appl. Logic 45 (1989), 277-308. MR 1032833 (91f:03068)
  • [5] S. Garavaglia, Decomposition of totally transcendental modules, J. Symbolic Logic 45 (1980), 155-164. MR 560233 (81a:03032)
  • [6] S. Givant, Universal Horn classes categorical or free in power, Ann. Math. Logic 15 (1978), 1-53. MR 511942 (80c:03032)
  • [7] -, A representation theorem for universal Horn classes categorical in power, Ann. Math. Logic 17 (1979), 91-116. MR 552417 (81b:03038)
  • [8] L. Harrington and M. Makkai, The main gap: Counting uncountable models of $ \omega $-stable and superstable theories, Notre Dame J. Formal Logic 26 (1985), 139-177. MR 783594 (87c:03075)
  • [9] C. Herrmann, Affine algebras in congruence modular varieties, Acta. Sci. Math. 41 (1979), 119-125. MR 534504 (80h:08011)
  • [10] B. Hart and S. Starchenko, Addendum to "A structure theorem for strongly abelian varieties", J. Symbolic Logic (to appear). MR 1253930 (95c:03073)
  • [11] B. Hart and M. Valeriote, A structure theorem for strongly abelian varieties with few models, J. Symbolic Logic 56 (1991), 832-852. MR 1129148 (93a:03031)
  • [12] Bradd Hart, An exposition of OTOP, Classification Theory: Chicago, 1985 (J. Baldwin, ed.), Lecture Notes in Math., vol. 1292, Springer-Verlag, Berlin and New York, 1987. MR 1033025 (90m:03069)
  • [13] D. Hobby and R. McKenzie, The structure of finite algebras, Contemp. Math., vol. 76, Amer. Math. Soc., Providence, R.I., 1989. MR 958685 (89m:08001)
  • [14] D. Lascar, Quelques précisions sur la DOP et la profondeur d'une théorie, J. Symbolic Logic 50 (1985), 316-330. MR 793109 (87a:03064)
  • [15] R. McKenzie, Categorical quasivarieties revisited, Algebra Universalis 19 (1984), 273-303. MR 779145 (87g:08022)
  • [16] R. McKenzie and M. Valeriote, The structure of locally finite varieties, vol. 78, Progress in Math., Birkhäuser, Boston, Mass., 1989. MR 1033992 (92j:08001)
  • [17] T. Mustafin, The stability theory of polygons, Akad. Nauk SSSR Sibirsk. Otdel. Trudy Inst. Mat. 8 (1988), 92-108. MR 957500 (90b:03047)
  • [18] E. Palyutin, Categorical Horn classes 1, Algebra i Logika 12 (1980). MR 566774 (81f:03042)
  • [19] -, The description of categorical quasivarieties, Algebra and Logic 14 (1976), 86-111.
  • [20] -, Spectra of varieties, Soviet Math. Dokl. 39 (1989), 553-554. MR 1014745 (90i:03035)
  • [21] E. Palyutin and S. Starchenko, Spectra of Horn classes, Soviet Math. Dokl. 34 (1987), 394-396. MR 866207 (88a:03079)
  • [22] R. Quackenbush, Quasi-affine algebras, Algebra Universalis 20 (1985), 318-327. MR 811692 (87d:08006)
  • [23] S. Shelah, Classification theory and the number of non-isomorphic models, 2nd ed., North-Holland, 1990. MR 1083551 (91k:03085)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03C45, 03C05, 03C60, 08B99

Retrieve articles in all journals with MSC: 03C45, 03C05, 03C60, 08B99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1191612-0
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society