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
DOI:
https://doi.org/10.1090/S0002-9939-1989-0975648-1
MathSciNet review:
975648
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a type bounded by an infinite regular cardinal
,
be a variety in
and
the class of all
-reducts of the algebras in
. We show that the operations in
are explicitely definable in
by pure formulas (i.e. existential-positive without disjunction) if and only if they are implicitely definable and
is closed under unions of
-chains (if and only if every
-homomorphisms between algebras in
are
-homomorphisms, as J. Isbell has shown). It follows that the operations in
are equivalent (in
) to
-terms if and only if every algebra in the
variety generated by
has a unique
-expansion in
.
- [1] Stanley Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics, vol. 78, Springer-Verlag, New York-Berlin, 1981. MR 648287
- [2] C. C. Chang and H. J. Keisler, Model theory, 2nd ed., North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Studies in Logic and the Foundations of Mathematics, 73. MR 0532927
- [3] K. L. de Bouvère, A mathematical characterization of explicit definability, Nederl. Akad. Wetensch. Proc. Ser. A 66=Indag. Math. 25 (1963), 264–274. MR 0156784
- [4] -, Synonymous theories, in Symposium on the Theory of Models, North-Holland, Amsterdam, 1965.
- [5] Peter Gabriel and Friedrich Ulmer, Lokal präsentierbare Kategorien, Lecture Notes in Mathematics, Vol. 221, Springer-Verlag, Berlin-New York, 1971 (German). MR 0327863
- [6] Michel Hébert, On the fullness of certain functors, J. Pure Appl. Algebra 61 (1989), no. 2, 181–188. MR 1025921, https://doi.org/10.1016/0022-4049(89)90012-1
- [7] John R. Isbell, Functorial implicit operations, Israel J. Math. 15 (1973), 185–188. MR 323671, https://doi.org/10.1007/BF02764604
- [8] R. N. McKenzie, Letter to S. Givant, 1983.
- [9] Hugo Volger, Preservation theorems for limits of structures and global sections of sheaves of structures, Math. Z. 166 (1979), no. 1, 27–54. MR 526864, https://doi.org/10.1007/BF01173845
- [10] G. E. Wever, Equational definability, manuscript, March 1987.
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
DOI:
https://doi.org/10.1090/S0002-9939-1989-0975648-1
Article copyright:
© Copyright 1989
American Mathematical Society