Sheaf constructions and their elementary properties
HTML articles powered by AMS MathViewer
- by Stanley Burris and Heinrich Werner PDF
- Trans. Amer. Math. Soc. 248 (1979), 269-309 Request permission
Abstract:
We are interested in sheaf constructions in model-theory, so an attempt is made to unify and generalize the results to date, namely various forms of the Feferman-Vaught Theorem, positive decidability results, and constructions of model companions. The task is considerably simplified by introducing a new definition of sheaf constructions over Boolean spaces.References
- Richard F. Arens and Irving Kaplansky, Topological representation of algebras, Trans. Amer. Math. Soc. 63 (1948), 457–481. MR 25453, DOI 10.1090/S0002-9947-1948-0025453-6
- C. J. Ash, Reduced powers and Boolean extensions, J. London Math. Soc. (2) 9 (1974/75), 429–432. MR 491144, DOI 10.1112/jlms/s2-9.3.429
- J. T. Baldwin and A. H. Lachlan, On universal Horn classes categorical in some infinte power, Algebra Universalis 3 (1973), 98–111. MR 351785, DOI 10.1007/BF02945108
- Bernhard Banaschewski and Evelyn Nelson, Boolean powers as algebras of continuous functions, Proceedings of the Lattice Theory Conference (Ulm, 1975) Univ. Ulm, Ulm, 1975, pp. 138–145. MR 0429693
- S. Bulman-Fleming and H. Werner, Equational compactness in quasi-primal varieties, Proceedings of the Lattice Theory Conference (Ulm, 1975) Univ. Ulm, Ulm, 1975, pp. 267–287. MR 0429695
- Stanley Burris, Boolean powers, Algebra Universalis 5 (1975), no. 3, 341–360. MR 447078, DOI 10.1007/BF02485268
- Stanley Burris and Heinrich Werner, Sheaf constructions and their elementary properties, Trans. Amer. Math. Soc. 248 (1979), no. 2, 269–309. MR 522263, DOI 10.1090/S0002-9947-1979-0522263-8 —, Decidable theories, Notices Amer. Math. Soc. 22 (1975), A-475. —, Model companions for discriminator varieties, Notices Amer. Math. Soc. 23 (1976), A-28. C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.
- Stephen D. Comer, Representations by algebras of sections over Boolean spaces, Pacific J. Math. 38 (1971), 29–38. MR 304277
- Stephen D. Comer, Elementary properties of structures of sections, Bol. Soc. Mat. Mexicana (2) 19 (1974), no. 2, 78–85. MR 437333
- Stephen D. Comer, Monadic algebras with finite degree, Algebra Universalis 5 (1975), no. 3, 313–327. MR 403965, DOI 10.1007/BF02485264
- Stephen D. Comer, Complete and model-complete theories of monadic algebras, Colloq. Math. 34 (1975/76), no. 2, 183–190. MR 480008, DOI 10.4064/cm-34-2-183-190
- Brian A. Davey, Sheaf spaces and sheaves of universal algebras, Math. Z. 134 (1973), 275–290. MR 330006, DOI 10.1007/BF01214692 A. Ehrenfeucht, Decidability of the theory of one function, Notices Amer. Math. Soc. 6 (1959), 268.
- Paul Eklof and Gabriel Sabbagh, Model-completions and modules, Ann. Math. Logic 2 (1970/71), no. 3, 251–295. MR 277372, DOI 10.1016/0003-4843(71)90016-7
- Ju. L. Eršov, Decidability of the elementary theory of relatively complemented lattices and of the theory of filters, Algebra i Logika Sem. 3 (1964), no. 3, 17–38 (Russian). MR 0180490
- Ju. L. Eršov, Elementary theories of Post varieties, Algebra i Logika 6 (1967), no. 5, 7–15 (Russian). MR 0231706
- Ju. L. Eršov, I. A. Lavrov, A. D. Taĭmanov, and M. A. Taĭclin, Elementary theories, Uspehi Mat. Nauk 20 (1965), no. 4 (124), 37–108 (Russian). MR 0186553
- S. Feferman and R. L. Vaught, The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57–103. MR 108455, DOI 10.4064/fm-47-1-57-103
- Paul R. Halmos, Lectures on Boolean algebras, Van Nostrand Mathematical Studies, No. 1, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. MR 0167440
- Leon Henkin, J. Donald Monk, and Alfred Tarski, Cylindric algebras. Part I. With an introductory chapter: General theory of algebras, Studies in Logic and the Foundations of Mathematics, Vol. 64, North-Holland Publishing Co., Amsterdam-London, 1971. MR 0314620
- Klaus Keimel and Heinrich Werner, Stone duality for varieties generated by quasi-primal algebras, Recent advances in the representation theory of rings and $C^{\ast }$-algebras by continuous sections (Sem., Tulane Univ., New Orleans, La., 1973), Mem. Amer. Math. Soc., No. 148, Amer. Math. Soc., Providence, R.I., 1974, pp. 59–85. MR 0360411
- Per Lindström, On model-completeness, Theoria (Lund) 30 (1964), 183–196. MR 179066, DOI 10.1111/j.1755-2567.1964.tb01088.x
- L. Lipshitz and D. Saracino, The model companion of the theory of commutative rings without nilpotent elements, Proc. Amer. Math. Soc. 38 (1973), 381–387. MR 439624, DOI 10.1090/S0002-9939-1973-0439624-8
- Angus Macintyre, Model-completeness for sheaves of structures, Fund. Math. 81 (1973/74), no. 1, 73–89. MR 337592, DOI 10.4064/fm-81-1-73-89
- J. A. Makowsky, On some conjectures connected with complete sentences, Fund. Math. 81 (1974), 193–202. MR 366647, DOI 10.4064/fm-81-3-193-202
- Anatoliĭ Ivanovič Mal′cev, The metamathematics of algebraic systems. Collected papers: 1936–1967, Studies in Logic and the Foundations of Mathematics, Vol. 66, North-Holland Publishing Co., Amsterdam-London, 1971. Translated, edited, and provided with supplementary notes by Benjamin Franklin Wells, III. MR 0349383
- R. S. Pierce, Modules over commutative regular rings, Memoirs of the American Mathematical Society, No. 70, American Mathematical Society, Providence, R.I., 1967. MR 0217056
- Michael O. Rabin, Decidability of second-order theories and automata on infinite trees, Trans. Amer. Math. Soc. 141 (1969), 1–35. MR 246760, DOI 10.1090/S0002-9947-1969-0246760-1 M. Rubin, Boolean algebras: Undecidability with distinguished subalgebra and interpretation in automorphism groups, Notices Amer. Math. Soc. 22 (1975), A-645.
- D. Saracino, Model companions for $\aleph _{0}$-categorical theories, Proc. Amer. Math. Soc. 39 (1973), 591–598. MR 316238, DOI 10.1090/S0002-9939-1973-0316238-1
- W. Szmielew, Elementary properties of Abelian groups, Fund. Math. 41 (1955), 203–271. MR 72131, DOI 10.4064/fm-41-2-203-271 A. Tarski, Arithmetical classes and types of Boolean algebras, Bull. Amer. Math. Soc. 55 (1949), 64.
- Walter Taylor, Pure compactifications in quasi-primal varieties, Canadian J. Math. 28 (1976), no. 1, 50–62. MR 401608, DOI 10.4153/CJM-1976-006-0
- Hugo Volger, The Feferman-Vaught theorem revisited, Colloq. Math. 36 (1976), no. 1, 1–11. MR 439619, DOI 10.4064/cm-36-1-1-11
- J. Waszkiewicz and B. Węglorz, Some models of theories of reduced powers, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 683–685 (English, with Russian summary). MR 237312
- J. Waszkiewicz and B. Węglorz, On $\omega _{0}$-categoricity of powers, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 195–199 (English, with Russian summary). MR 258616
- B. Węglorz, Substructures of reduced powers, Fund. Math. 89 (1975), no. 3, 191–197. MR 414351, DOI 10.4064/fm-89-3-191-197
- Volker Weispfenning, Model-completeness and elimination of quantifiers for subdirect products of structures, J. Algebra 36 (1975), no. 2, 252–277. MR 437339, DOI 10.1016/0021-8693(75)90101-5
- Heinrich Werner, Discriminator-algebras, Studien zur Algebra und ihre Anwendungen [Studies in Algebra and its Applications], vol. 6, Akademie-Verlag, Berlin, 1978. Algebraic representation and model theoretic properties. MR 526402
- Andrew B. Carson, The model completion of the theory of commutative regular rings, J. Algebra 27 (1973), 136–146. MR 387046, DOI 10.1016/0021-8693(73)90169-5
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 248 (1979), 269-309
- MSC: Primary 03C30; Secondary 03C90
- DOI: https://doi.org/10.1090/S0002-9947-1979-0522263-8
- MathSciNet review: 522263