Compatible idempotent terms in universal algebra

In universal algebra, we oftentimes encounter varieties that are not especially well-behaved from any point of view, but are such that all their members have a well-behaved core, i.e. subalgebras or quotients with satisfactory properties. Of special interest is the case in which this coreis a retract determined by an idempotent endomorphism that is uniformly term denable (through a unary term t(x)) in every member of the given variety. Here, we try to give a unied account of this phenomenon. In particular, we investigate what happens when various congruence properties like congruence distributivity, congruence permutability or congruence modularity are not supposed to hold unrestrictedly in any A 2 V, but only for congruence classes of values of the term operation tA.