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Proceedings of the American Mathematical Society

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

 

 

Two observations on the congruence extension property


Authors: G. Grätzer and H. Lakser
Journal: Proc. Amer. Math. Soc. 35 (1972), 63-64
MSC: Primary 08A25
DOI: https://doi.org/10.1090/S0002-9939-1972-0297677-3
MathSciNet review: 0297677
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Abstract: A pair of algebras $ \mathfrak{U},\mathfrak{B}$ with $ \mathfrak{B}$ a subalgebra of $ \mathfrak{U}$ is said to have the (Principal) Congruence Extension Property (abbreviated as PCEP and CEP, respectively) if every (principal) congruence relation of $ \mathfrak{B}$ can be extended to $ \mathfrak{U}$. A pair of algebras $ \mathfrak{U}$, $ \mathfrak{B}$ is constructed having PCEP but not CEP, solving a problem of A. Day. A result of A. Day states that if $ \mathfrak{B}$ is a subalgebra of $ \mathfrak{U}$ and if for any subalgebra $ \mathfrak{C}$ of $ \mathfrak{U}$ containing $ \mathfrak{B}$, the pair $ \mathfrak{U},\mathfrak{C}$ has PCEP, then $ \mathfrak{U},\mathfrak{B}$ has CEP. A new proof of this theorem that avoids the use of the Axiom of Choice is also given.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0297677-3
Keywords: Universal algebra, congruence relation, Congruence Extension Property
Article copyright: © Copyright 1972 American Mathematical Society