Generic algebras

Isbell, John

doi:10.1090/S0002-9947-1983-0682715-8

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.

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