Abstract
The question whether the extended theorem of Feferman and Vaught (cf. Comer [7], Volger [13]) can be applied to filtered boolean powers with arbitrary base was raised by J. H. Schmerl [11]. It can be answered with the help of a relativized version of the theorem (cf. Comer [7], Volger [12], [14]) which seems to have escaped notice. Therefore we shall present in this note a precise formulation of the relativized version and apply it to filtered boolean powers with arbitrary base and to stable boolean powers with finite base and finite group. The former application was mentioned in Comer [7]. The latter application yields a decidability result which seems to be new. The question whether the second application can be extended to arbitrary bases remains open.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. F. Arens andI. Kaplansky,Topological representations of algebras, Trans. Amer. Math. Soc.63 (1948), 457–481.
B.Banaschewski and E.Nelson,Boolean powers as algebras of continuous sections, Diss. Math.179 (1980), 51 pp.
S. Burris,The first order theory of boolean algebras with a distinguished group of automorphisms, Algebra Universalis15 (1982), 156–161.
S. Burris, andJ. Lawrence,Two undecidability results using modified boolean powers, Can. J. Math.34 (1982), 500–505.
S. Burris andH. Werner,Sheaf constructions and their elementary properties, Trans. Amer. Math. Soc.248 (1979), 269–309.
S. D. Comer,Representations by algebras of sections over boolean spaces, Pacific J. Math.38 (1971), 29–38.
S. D. Comer,Elementary properties of structures of sections, Bol. Soc. Math. Mexicana (2)19 (1974), 78–85.
S. Feferman andR. L. Vaught,The first-order properties of products of algebraic systems, Fund. Math.47 (1959), 57–103.
P. H.Krauss and D. M.Clark,Global subdirect products, Memoirs Amer. Math. Soc.210 (1979), 109 pp.
F.Miraglia,Relations between structures of stable continuous functions and filtered powers, in: Proc. 3rd Brazilian Conf. on Math. Logic, A. I. Arruda, A. M. Sette, N. C. A. da Costa, eds. (1980), 225–244.
J. H. Schmerl,On ℵ 0-categoricity of filtered boolean extensions, Algebra Universalis8 (1978), 159–161.
H.Volger,The Feferman-Vaught theorem revisited, Tagungsbericht 4/1976, Model theory Conf. Jan. 1976.
H. Volger,The Feferman-Vaught theorem revisited, Colloq. Math.36 (1976), 1–11.
H. Volger,Sheaf representations of algebras and transfer theorems for model theoretic properties, Studien zur Algebra und ihre Anwendungen 7, Akademie Verlag, Berlin 1979, 155–163.
J. Waskiewicz,ω 0-categoricity of generalized products, Colloq. Math.27 (1973), 1–5.
H. Werner,Discriminator algebras, Studien zur Algebra und ihre Anwendungen 6, Akademie Verlag, Berlin 1978.
V. Weispfenning,Model theory of lattice products, Habilitationsschrift, Univ. Heidelberg, 1978.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Volger, H. Filtered and stable boolean powers are relativized full boolean powers. Algebra Universalis 19, 399–402 (1984). https://doi.org/10.1007/BF01201109
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01201109