An algebra $$\mathbf{A}$$ is called a perfect extension of its subalgebra $$\mathbf{B}$$ if every congruence has unique to . This terminology was used by Blyth and Varlet [1994]. In the case lattices, this concept described Grätzer Wehrung [1999] saying that congruence-preserving Not many investigations have been carried out so far. The present authors in another recent study faced question whe...

Recent investigations of lax algebras—in generalization of Barr’s relational algebras—make an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and V-relations (where V is a unital quantale). For a given monad there may be many such lax extensions, and different constructions appear in the literature. The aim of this article is to shed a unifying light on t...