## 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