Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Generic algebras
HTML articles powered by AMS MathViewer

by John Isbell PDF
Trans. Amer. Math. Soc. 275 (1983), 497-510 Request permission

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
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 18C05, 08B99
  • Retrieve articles in all journals with MSC: 18C05, 08B99
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