We provide a Maltsev characterization of congruence distributive varieties by showing that a variety 𝓥 is congruence distributive if and only if the congruence identity … (k factors) holds in 𝓥, for some natural number k.
The relationship between absolute retracts, injectives and equationally compact algebras in finitely generated congruence
distributive varieties with 1- element subalgebras is considered and several characterization theorems are proven. Amongst
others, we prove that the absolute retracts in such a variety are precisely the injectives in the amalgamation class and that
every equationally compact reduced power of a finite absolute retract is an absolute retract. We also show that any elementary
amalgamation class is Horn if and only if it is closed under finite direct products.