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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Generic algebras

Author: John Isbell
Journal: Trans. Amer. Math. Soc. 275 (1983), 497-510
MSC: Primary 18C05; Secondary 08B99
Erratum: Trans. Amer. Math. Soc. 295 (1986), 429.
MathSciNet review: 682715
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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.

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Keywords: Functorial semantics, variety, epimorphism, definable
Article copyright: © Copyright 1983 American Mathematical Society