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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Two definability results in the equational context


Authors: M. Hébert, R. N. McKenzie and G. E. Weaver
Journal: Proc. Amer. Math. Soc. 107 (1989), 47-53
MSC: Primary 08B05; Secondary 03C05, 03C40
MathSciNet review: 975648
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Abstract: Let $ \tau $ be a type bounded by an infinite regular cardinal $ \alpha $, $ {\mathbf{V}}$ be a variety in $ \tau ,\tau ' \subseteq \tau $ and $ {\mathbf{V'}}$ the class of all $ \tau '$-reducts of the algebras in $ {\mathbf{V}}$. We show that the operations in $ \tau \backslash \tau '$ are explicitely definable in $ {\mathbf{V}}$ by pure formulas (i.e. existential-positive without disjunction) if and only if they are implicitely definable and $ {\mathbf{V'}}$ is closed under unions of $ \alpha $-chains (if and only if every $ \tau '$-homomorphisms between algebras in $ {\mathbf{V}}$ are $ \tau $-homomorphisms, as J. Isbell has shown). It follows that the operations in $ \tau \backslash \tau '$ are equivalent (in $ {\mathbf{V}}$) to $ \tau '$-terms if and only if every algebra in the $ (\tau ' - )$ variety generated by $ {\mathbf{V'}}$ has a unique $ \tau $-expansion in $ {\mathbf{V}}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0975648-1
PII: S 0002-9939(1989)0975648-1
Article copyright: © Copyright 1989 American Mathematical Society