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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.

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The Fraser-Horn and Apple properties
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by Joel Berman and W. J. Blok PDF
Trans. Amer. Math. Soc. 302 (1987), 427-465 Request permission


We consider varieties $\mathcal {V}$ in which finite direct products are skew-free and in which the congruence lattices of finite directly indecomposables have a unique coatom. We associate with $\mathcal {V}$ a family of derived varieties, $d(\mathcal {V})$: a variety in $d(\mathcal {V})$ is generated by algebras ${\mathbf {A}}$ where the universe of ${\mathbf {A}}$ consists of a congruence class of the coatomic congruence of a finite directly indecomposable algebra ${\mathbf {B}} \in \mathcal {V}$ and the operations of ${\mathbf {A}}$ are those of ${\mathbf {B}}$ that preserve this congruence class. We also consider the prime variety of $\mathcal {V}$, denoted ${\mathcal {V}_0}$, generated by all finite simple algebras in $\mathcal {V}$. We show how the structure of finite algebras in $\mathcal {V}$ is determined to a considerable extent by ${\mathcal {V}_0}$ and $d(\mathcal {V})$. In particular, the free $\mathcal {V}$-algebra on $n$ generators, ${{\mathbf {F}}_\mathcal {V}}(n)$, has as many directly indecomposable factors as ${{\mathbf {F}}_{{\mathcal {V}_0}}}(n)$ and the structure of these factors is determined by the varieties $d(\mathcal {V})$. This allows us to produce in many cases explicit formulas for the cardinality of ${{\mathbf {F}}_\mathcal {V}}(n)$. Our work generalizes the structure theory of discriminator varieties and, more generally, that of arithmetical semisimple varieties. The paper contains many examples of algebraic systems that have been investigated in different contexts; we show how these all fit into a general scheme.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 302 (1987), 427-465
  • MSC: Primary 08B20; Secondary 03G25, 08A40
  • DOI:
  • MathSciNet review: 891629