On the behaviour of congruence-functors
Let Comp \({\mathfrak A}\) be the semilattice of all compact congruences of the algebra \({\mathfrak A}\). Any homomorphisms \(\phi\) : \({\mathfrak A}\to {\mathfrak B}\) of algebras of a variety K induces a homomorphism Comp \(\phi\) by the identity (Comp \(\phi)\)(\(\Theta)\)(a,a')\(=\Theta (\phi (a),\phi (a'))\). It is proved that the functor Comp has the following properties: it preserves products iff the principal congruences can be defined in K by a finite set of equations; it always preserves surjectivity; it preserves injectivity iff K satisfies the Congruence Extension Property; it preserves products and injectivity iff the equation above contains only the four elements which are involved in the congruence; it never preserves coproducts.
scientific article; zbMATH DE number 4051704