Geodesic
Two Undecidability Results using Modified Boolean Powers
Canadian journal of mathematics, Tome 34 (1982) no. 2, pp. 500-505

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper we will give brief proofs of two results on the undecidability of a first-order theory using a construction which we call a modified Boolean power. Modified Boolean powers were introduced by Burris in late 1978, and the first results were announced in [2]. Subsequently we succeeded in using this construction to prove the results in this paper, namely Ershov's theorem that every variety of groups containing a finite non-abelian group has an undecidable theory, and Zamjatin's theorem that a variety of rings with unity which is not generated by finitely many finite fields has an undecidable theory. Later McKenzie further modified the construction mentioned above, and combined it with a variant of one of Zamjatin's constructions to prove the sweeping main result of [3]. The proofs given here have the advantage (over the original proofs) that they use a single construction. 
Burris, Stanley; Lawrence, John. Two Undecidability Results using Modified Boolean Powers. Canadian journal of mathematics, Tome 34 (1982) no. 2, pp. 500-505. doi: 10.4153/CJM-1982-033-6
@article{10_4153_CJM_1982_033_6,
     author = {Burris, Stanley and Lawrence, John},
     title = {Two {Undecidability} {Results} using {Modified} {Boolean} {Powers}},
     journal = {Canadian journal of mathematics},
     pages = {500--505},
     year = {1982},
     volume = {34},
     number = {2},
     doi = {10.4153/CJM-1982-033-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-033-6/}
}
TY  - JOUR
AU  - Burris, Stanley
AU  - Lawrence, John
TI  - Two Undecidability Results using Modified Boolean Powers
JO  - Canadian journal of mathematics
PY  - 1982
SP  - 500
EP  - 505
VL  - 34
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-033-6/
DO  - 10.4153/CJM-1982-033-6
ID  - 10_4153_CJM_1982_033_6
ER  - 
%0 Journal Article
%A Burris, Stanley
%A Lawrence, John
%T Two Undecidability Results using Modified Boolean Powers
%J Canadian journal of mathematics
%D 1982
%P 500-505
%V 34
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-033-6/
%R 10.4153/CJM-1982-033-6
%F 10_4153_CJM_1982_033_6

[1] 1. Arens, R. F. and Kaplansky, I., Topological representations of algebras, Trans. Amer. Math. Soc. 63 (1948), 457–481. Google Scholar

[2] 2. Burris, S., An algebraic test for undecidability, N.A.M.S. 26 (1979). Google Scholar

[3] 3. Burris, S. and McKenzie, R., Decidability and Boolean representations, Memoirs Amer. Math. Soc. 32, No. 246 (1981). Google Scholar

[4] 4. Comer, S., Elementary properties of structures of sections, Bol. Soc. Mat. Mexicana 19 (1974), 78–85. Google Scholar

[5] 5. Yu. L., Ersov, Theories of non-abelian varieties of groups, Proc. of Symposium in Pure Mathematics 25 (A.M.S., Providence, Rhode Island, 1974), 255–264. Google Scholar

[6] 6. Scott, W. R., Group theory (Prentice Hall, 1964). Google Scholar

[7] 7. Zamjatin, A. P., Varieties of associative rings whose elementary theory is decidable, Soviet Math. Doklady 17 (1976), 996–999. Google Scholar

Cité par Sources :