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Pseudobases in direct powers of an algebra


Author: Paul Bankston
Journal: Trans. Amer. Math. Soc. 335 (1993), 79-90
MSC: Primary 08A35; Secondary 03C05, 12L10
DOI: https://doi.org/10.1090/S0002-9947-1993-1155348-3
MathSciNet review: 1155348
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Abstract: A subset $ P$ of an abstract algebra $ A$ is a pseudobasis if every function from $ P$ into $ A$ extends uniquely to an endomorphism on $ A$. $ A$ is called $ \kappa $-free has a pseudobasis of cardinality $ \kappa $; $ A$ is minimally free if $ A$ has a pseudobasis. (The 0-free algebras are "rigid" in the strong sense; the $ 1$-free groups are always abelian, and are precisely the additive groups of $ E$-rings.) Our interest here is in the existence of pseudobases in direct powers $ {A^I}$ of an algebra $ A$. On the positive side, if $ A$ is a rigid division ring, $ \kappa $ is a cardinal, and there is no measurable cardinal $ \mu $ with $ \vert A\vert < \mu \leq \kappa $, then $ {A^I}$ is $ \kappa $-free whenever $ \vert I\vert = \vert{A^\kappa }\vert$. On the negative side, if $ A$ is a rigid division ring and there is a measurable cardinal $ \mu $ with $ \vert A\vert < \mu \leq \vert I\vert$, then $ {A^I}$ is not minimally free.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1155348-3
Keywords: Pseudobases, direct powers, universal algebra, minimally free, fields, division rings, measurable cardinals
Article copyright: © Copyright 1993 American Mathematical Society

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