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 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**, 10.1016/0022-4049(89)90012-1**[7]**John R. Isbell,*Functorial implicit operations*, Israel J. Math.**15**(1973), 185–188. MR**0323671****[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**, 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