Generic algebras
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- by John Isbell PDF
- Trans. Amer. Math. Soc. 275 (1983), 497-510 Request permission
Erratum: Trans. Amer. Math. Soc. 295 (1986), 429.
Abstract:
The familiar (merely) generic algebras in a variety $\mathcal {V}$ are those which separate all the different operations of $\mathcal {V}$, or equivalently lie in no proper Birkhoff subcategory. Stronger notions are considered, the strongest being canonicalness of a (small) subcategory $\mathcal {A}$ of $\mathcal {V}$, defined: the structure functor takes inclusion $\mathcal {A} \subset \mathcal {V}$ to an isomorphism of varietal theories. Intermediate are dominance and exemplariness: lying in no proper varietal subcategory, respectively full subcategory. It is shown that, modulo measurable cardinals, every finitary variety has a canonical set (subcategory) of one or two algebras, the possible second one being the empty algebra. Without reservation, every variety with rank has a dominant set of one or two algebras (the second as before). Finally, in modules over a ring $R$, the generic module $R$ is shown to be (a) dominant if exemplary, and (b) dominant if $R$ is countable or right artinian. However, power series rings $R$ and some others are not dominant $R$-modules.References
- Andrzej Ehrenfeucht, Siemion Fajtlowicz, and Jan Mycielski, Homomorphisms of direct powers of algebras, Fund. Math. 103 (1979), no. 3, 189–203. MR 547553, DOI 10.4064/fm-103-3-189-203
- Otto Gerstner, Ludger Kaup, and Hans-Günther Weidner, Whitehead-Moduln abzählbaren Ranges über Hauptidealringen, Arch. Math. (Basel) 20 (1969), 503–514 (German). MR 252379, DOI 10.1007/BF01899457
- Wilfrid Hodges and Saharon Shelah, Infinite games and reduced products, Ann. Math. Logic 20 (1981), no. 1, 77–108. MR 611395, DOI 10.1016/0003-4843(81)90012-7
- J. R. Isbell, Adequate subcategories, Illinois J. Math. 4 (1960), 541–552. MR 175954
- J. R. Isbell, Subjects, adequacy, completeness and categories of algebras. [Subobjects, adequacy, completeness and categories of algebras], Rozprawy Mat. 36 (1964), 33. MR 163939
- John R. Isbell, Epimorphisms and dominions, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 232–246. MR 0209202
- John R. Isbell, Epimorphisms and dominions. III, Amer. J. Math. 90 (1968), 1025–1030. MR 237596, DOI 10.2307/2373286
- John R. Isbell, Functorial implicit operations, Israel J. Math. 15 (1973), 185–188. MR 323671, DOI 10.1007/BF02764604
- C. U. Jensen, Les foncteurs dérivés de $\underleftarrow {\mmlToken {mi}{lim}}$ et leurs applications en théorie des modules, Lecture Notes in Mathematics, Vol. 254, Springer-Verlag, Berlin-New York, 1972. MR 0407091
- J. F. Kennison and Dion Gildenhuys, Equational completion, model induced triples and pro-objects, J. Pure Appl. Algebra 1 (1971), no. 4, 317–346. MR 306289, DOI 10.1016/0022-4049(71)90001-6
- F. William Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 869–872. MR 158921, DOI 10.1073/pnas.50.5.869
- F. E. J. Linton, Some aspects of equational categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 84–94. MR 0209335
- Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
- Ernest G. Manes, Algebraic theories, Graduate Texts in Mathematics, No. 26, Springer-Verlag, New York-Heidelberg, 1976. MR 0419557
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 275 (1983), 497-510
- MSC: Primary 18C05; Secondary 08B99
- DOI: https://doi.org/10.1090/S0002-9947-1983-0682715-8
- MathSciNet review: 682715