Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Two definability results in the equational context

Authors: M. Hébert, R. N. McKenzie and G. E. Weaver
Journal: Proc. Amer. Math. Soc. 107 (1989), 47-53
MSC: Primary 08B05; Secondary 03C05, 03C40
MathSciNet review: 975648
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \tau $ be a type bounded by an infinite regular cardinal $ \alpha $, $ {\mathbf{V}}$ be a variety in $ \tau ,\tau ' \subseteq \tau $ and $ {\mathbf{V'}}$ the class of all $ \tau '$-reducts of the algebras in $ {\mathbf{V}}$. We show that the operations in $ \tau \backslash \tau '$ are explicitely definable in $ {\mathbf{V}}$ by pure formulas (i.e. existential-positive without disjunction) if and only if they are implicitely definable and $ {\mathbf{V'}}$ is closed under unions of $ \alpha $-chains (if and only if every $ \tau '$-homomorphisms between algebras in $ {\mathbf{V}}$ are $ \tau $-homomorphisms, as J. Isbell has shown). It follows that the operations in $ \tau \backslash \tau '$ are equivalent (in $ {\mathbf{V}}$) to $ \tau '$-terms if and only if every algebra in the $ (\tau ' - )$ variety generated by $ {\mathbf{V'}}$ has a unique $ \tau $-expansion in $ {\mathbf{V}}$.

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

  • [1] S. Burris and H. P. Sankappanavar, A course in universal algebra, Springer-Verlag, New York, 1980. MR 648287 (83k:08001)
  • [2] C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1977. MR 0532927 (58:27177)
  • [3] K. L. De Bouvère, A mathematical characterization of explicit definability, Indag. Math. 25 (1963), 264-274. MR 0156784 (28:28)
  • [4] -, Synonymous theories, in Symposium on the Theory of Models, North-Holland, Amsterdam, 1965.
  • [5] P. Gabriel and F. Ulmer, Lokal prasentierbare Kategorien, Lecture Notes in Math., no. 221, Springer-Verlag, Berlin, 1971. MR 0327863 (48:6205)
  • [6] M. Hébert, On the fullness of certain functors, (to appear in J. Pure Appl. Algebra). MR 1025921 (90j:18001)
  • [7] J. R. Isbell, Functorial implicit operations, Israel J. Math. 15 (1973), 185-188. MR 0323671 (48:2027)
  • [8] R. N. McKenzie, Letter to S. Givant, 1983.
  • [9] H. Volger, Preservation theorems for limits of structures and global sections of sheaves of structures, Math. Z. 166 (1979), 27-53. MR 526864 (80d:03029)
  • [10] G. E. Wever, Equational definability, manuscript, March 1987.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 08B05, 03C05, 03C40

Retrieve articles in all journals with MSC: 08B05, 03C05, 03C40

Additional Information

Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society