AbstractThe D-compatible semigroup varieties will be characterized and described. It is found that a semigroup variety is D-compatible if and only if it is J-compatible. It is shown that a periodic semigroup variety contains at most six maximal D-compatible subvarieties and every D-compatible subvariety is contained in one of these maximal ones. The semigroup varieties which are minimal for not being D-compatible are found: they are all periodic and countably infinite in number. There are six distinct maximal D-compatible pseudovarieties of semigroups. The semigroup varieties and pseudovarieties which are compatible for each of the Green relations are characterized and described. Analogues for varieties and pseudovarieties of monoids are established. It is shown that if a D-compatible variety of monoids contains a nonabelian group, then it is periodic and consists of completely regular monoids only.