Two observations on the congruence extension property

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.