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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Extending congruence relations

Author: Peter Krauss
Journal: Proc. Amer. Math. Soc. 31 (1972), 517-520
MathSciNet review: 0285470
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Abstract: If $ \mathfrak{A}$ and $ \mathfrak{B}$ are algebras, where $ \mathfrak{A} \subseteq \mathfrak{B}$, and $ \theta $ is a congruence relation on $ \mathfrak{A}$, let $ {\theta ^\mathfrak{B}}$ be the smallest congruence relation on $ \mathfrak{B}$ containing $ \theta $. $ \mathfrak{A}$ is called a congruence subalgebra of $ \mathfrak{B}$ if $ \mathfrak{A} \subseteq \mathfrak{B}$ and, for every congruence relation $ \theta $ on $ \mathfrak{A},{\theta ^\mathfrak{B}} \cap \vert\mathfrak{A}{\vert^2} = \theta $. Elementary subalgebras are congruence subalgebras, and there are Directed Union and Loewenheim-Skolem Theorems for congruence subalgebras analogous to those for elementary subalgebras. Consequently we obtain full analogues of the Jónsson-Morley-Vaught results concerning homogeneous-universal algebras, where the notion of ``subalgebra'' is everywhere replaced by ``congruence subalgebra".

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Keywords: Congruence relation, elementary subalgebra, Directed Union Theorem, Loewenheim-Skolem Theorem, embedding property, amalgamation property, homogeneous-universal algebra
Article copyright: © Copyright 1972 American Mathematical Society

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