Abstract
A pair of algebras U , B \mathfrak {U},\mathfrak {B} with B \mathfrak {B} a subalgebra of U \mathfrak {U} is said to have the (Principal) Congruence Extension Property (abbreviated as PCEP and CEP, respectively) if every (principal) congruence relation of B \mathfrak {B} can be extended to U \mathfrak {U} . A pair of algebras U \mathfrak {U} , B \mathfrak {B} is constructed having PCEP but not CEP, solving a problem of A. Day. A result of A. Day states that if B \mathfrak {B} is a subalgebra of U \mathfrak {U} and if for any subalgebra C \mathfrak {C} of U \mathfrak {U} containing B \mathfrak {B} , the pair U , C \mathfrak {U},\mathfrak {C} has PCEP, then U , B \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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