Some universal sets of terms
HTML articles powered by AMS MathViewer
- by Walter Taylor PDF
- Trans. Amer. Math. Soc. 267 (1981), 595-607 Request permission
Abstract:
For every $\Pi _2^1$ class of cardinals containing $0$ and $1$, there exists a finite set $T$ of terms, such that $X$ is precisely the class of cardinals in which $T$ is universal.References
- J. W. Addison, Leon Henkin, and Alfred Tarski (eds.), The theory of models, North-Holland Publishing Co., Amsterdam, 1965. MR 0195680
- John T. Baldwin and Joel Berman, Elementary classes of varieties, Houston J. Math. 7 (1981), no. 4, 473–492. MR 658563 S. Banach, Sur un théorème de M. Sierpiński, Fund. Math. 25 (1935), 5-6. C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.
- W. Craig and R. L. Vaught, Finite axiomatizability using additional predicates, J. Symbolic Logic 23 (1958), 289–308. MR 106175, DOI 10.2307/2964289
- Andrzej Ehrenfeucht, Elementary theories with models without automorphisms, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 70–76. MR 0209133
- A. Ehrenfeucht, S. Fajtlowicz, J. Malitz, and J. Mycielski, Some problems on the universality of words in groups, Algebra Universalis 11 (1980), no. 2, 261–263. MR 588219, DOI 10.1007/BF02483104
- Andrzej Ehrenfeucht and D. M. Silberger, Decomposing a transformation with an involution, Algebra Universalis 7 (1977), no. 2, 179–190. MR 437341, DOI 10.1007/BF02485427
- A. Ehrenfeucht and D. M. Silberger, Universal and point universal terms, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), no. 6, 399–402 (English, with Russian summary). MR 409690
- Andrzej Ehrenfeucht and D. M. Silberger, Universal terms of the form $B^{n}A^{m}$, Algebra Universalis 10 (1980), no. 1, 96–116. MR 552160, DOI 10.1007/BF02482894
- P. Erdős, A. Hajnal, and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93–196. MR 202613, DOI 10.1007/BF01886396
- P. Erdös and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427–489. MR 81864, DOI 10.1090/S0002-9904-1956-10036-0
- Ralph Freese and Ralph McKenzie, Residually small varieties with modular congruence lattices, Trans. Amer. Math. Soc. 264 (1981), no. 2, 419–430. MR 603772, DOI 10.1090/S0002-9947-1981-0603772-9
- Stephen J. Garland, Second-order cardinal characterizability, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1974, pp. 127–146. MR 0416919
- J. R. Isbell, On the problem of universal terms, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 (1966), 593–595 (English, with Russian summary). MR 207567
- J. R. Isbell, M. I. Klun, and S. H. Schanuel, Affine parts of algebraic theories. I, J. Algebra 44 (1977), no. 1, 1–8. MR 429542, DOI 10.1016/0021-8693(77)90164-8
- B. Jónsson, The unique factorization problem for finite relational structures, Colloq. Math. 14 (1966), 1–32. MR 191860, DOI 10.4064/cm-14-1-1-32
- Bjarni Jónsson, Topics in universal algebra, Lecture Notes in Mathematics, Vol. 250, Springer-Verlag, Berlin-New York, 1972. MR 0345895
- S. C. Kleene, Finite axiomatizability of theories in the predicate calculus using additional predicate symbols. Two papers on the predicate calculus, Mem. Amer. Math. Soc. 10 (1952), 27–68. MR 51186 A. Kostinsky and R. McKenzie, Notices Amer. Math. Soc. 15 (1968), 389; Abstract #68T-302. A. Kostinsky, Some problems for rings and lattices within the domain of general algebra, Ph.D. Thesis, Berkeley, Calif., 1969.
- Kenneth Kunen, Indescribability and the continuum, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 199–203. MR 0282829
- W. A. Lampe and W. Taylor, Simple algebras in varieties, Algebra Universalis 14 (1982), no. 1, 36–43. MR 634414, DOI 10.1007/BF02483905
- James T. Loats, Hopfian Boolean algebras of power less than or equal to continuum, Proc. Amer. Math. Soc. 77 (1979), no. 2, 186–190. MR 542082, DOI 10.1090/S0002-9939-1979-0542082-1
- James Loats and Matatyahu Rubin, Boolean algebras without nontrivial onto endomorphisms exist in every uncountable cardinality, Proc. Amer. Math. Soc. 72 (1978), no. 2, 346–351. MR 507336, DOI 10.1090/S0002-9939-1978-0507336-2
- G. G. Lorentz, The 13th problem of Hilbert, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R.I., 1976, pp. 419–430. MR 0507425
- Jerzy Łoś, Un théorème sur les superpositions des fonctions définies dans les ensembles arbitraires, Fund. Math. 37 (1950), 84–86 (French). MR 41202, DOI 10.4064/fm-37-1-84-86
- Ralph McKenzie, On elementary types of symmetric groups, Algebra Universalis 1 (1971), no. 1, 13–20. MR 286643, DOI 10.1007/BF02944950
- Ralph McKenzie, On semigroups whose proper subsemigroups have lesser power, Algebra Universalis 1 (1971), no. 1, 21–25. MR 292979, DOI 10.1007/BF02944951
- Ralph McKenzie and Saharon Shelah, The cardinals of simple models for universal theories, Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1974, pp. 53–74. MR 0360261
- George F. McNulty, The decision problem for equational bases of algebras, Ann. Math. Logic 10 (1976), no. 3-4, 193–259. MR 432440, DOI 10.1016/0003-4843(76)90009-7
- Donald Monk, On pseudo-simple universal algebras, Proc. Amer. Math. Soc. 13 (1962), 543–546. MR 144840, DOI 10.1090/S0002-9939-1962-0144840-1
- Michael Morley, Omitting classes of elements, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 265–273. MR 0201305
- John Myhill and Andrew Adler, Problems and Solutions: Solutions of Advanced Problems: 6244, Amer. Math. Monthly 87 (1980), no. 8, 676–678. MR 1539504
- Peter Perkins, Unsolvable problems for equational theories, Notre Dame J. Formal Logic 8 (1967), 175–185. MR 236012
- A. G. Pinus, Elementary definability of symmetry groups, Algebra Universalis 3 (1973), 59–66. MR 337594, DOI 10.1007/BF02945104
- E. Redi, Representation of Menger systems by multiplace endomorphisms, Tartu Riikl. Ül. Toimetised 277 (1971), 47–51 (Russian, with Estonian and English summaries). MR 0371789
- Saharon Shelah, First order theory of permutation groups, Israel J. Math. 14 (1973), 149–162; errata, ibid. 15 (1973), 437–441. MR 416909, DOI 10.1007/BF02762670
- Saharon Shelah, Notes on combinatorial set theory, Israel J. Math. 14 (1973), 262–277. MR 327522, DOI 10.1007/BF02764885
- Saharon Shelah, Refuting Ehrenfeucht conjecture on rigid models, Israel J. Math. 25 (1976), no. 3-4, 273–286. MR 485326, DOI 10.1007/BF02757005 —, On a problem of Kurosh, Jónsson groups, and applications, Word Problems, Decision Problems and the Burnside Problem in Group Theory. II (Adjan, Boone and Higman, Editors) (to appear). W. Sierpiński, Sur les suites infinies de fonctions définies dans les ensembles quelconques, Fund. Math. 24 (1934), 209-212.
- D. M. Silberger, When is a term point universal?, Algebra Universalis 10 (1980), no. 2, 135–154. MR 560137, DOI 10.1007/BF02482898
- D. M. Silberger, Universal terms of complexity three, Algebra Universalis 11 (1980), no. 3, 393–395 (1981). MR 602024, DOI 10.1007/BF02483116
- D. M. Silberger, $B^{n}A^{m}$ is universal iff point universal, Algebra Universalis 12 (1981), no. 3, 335–342. MR 624299, DOI 10.1007/BF02483893
- Alfred Tarski, Undecidable theories, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1953. In collaboration with Andrzej Mostowski and Raphael M. Robinson. MR 0058532
- Walter Taylor, Residually small varieties, Algebra Universalis 2 (1972), 33–53. MR 314726, DOI 10.1007/BF02945005
- Walter Taylor, Equational logic, Houston J. Math. Survey (1979), iii+83. MR 546853
- Walter Taylor, Laws obeyed by topological algebras—extending results of Hopf and Adams, J. Pure Appl. Algebra 21 (1981), no. 1, 75–98. MR 609272, DOI 10.1016/0022-4049(81)90076-1 —, Abstracts Amer. Math. Soc. 1 (1980), 218, 390; Abstracts #80T-A61, #80T-E57. M. L. Valente, Sobre a univeralidade de palavras para grupos simétricos, MS Thesis, Florianópolis, 1979.
- A. G. Vitushkin, On representation of functions by means of superpositions and related topics, Enseign. Math. (2) 23 (1977), no. 3-4, 255–320. MR 499036
- Peter M. Winkler, Polynomial hyperforms, Algebra Universalis 17 (1983), no. 1, 101–109. MR 710002, DOI 10.1007/BF01194518
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 267 (1981), 595-607
- MSC: Primary 08A40; Secondary 03D35, 04A05
- DOI: https://doi.org/10.1090/S0002-9947-1981-0626492-3
- MathSciNet review: 626492