Extending congruence relations
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- by Peter Krauss PDF
- Proc. Amer. Math. Soc. 31 (1972), 517-520 Request permission
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 $\mathfrak {A},{\theta ^\mathfrak {B}} \cap |\mathfrak {A}{|^2} = \theta$. 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".References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 31 (1972), 517-520
- DOI: https://doi.org/10.1090/S0002-9939-1972-0285470-7
- MathSciNet review: 0285470