Abstract. The ${\cal R}$-compatible semigroup varieties are classified and described. It is shown that a periodic semigroup variety contains at most three maximal ${\cal R}$-compatible subvarieties, and each ${\cal R}$-compatible subvariety is contained in one of the maximal ones. The semigroup varieties which are minimal for not being ${\cal R}$-compatible are found: they are countably infinite in number. There are three maximal ${\cal R}$-compatible pseudovarieties of semigroups. Analogues for varieties and pseudovarieties of monoids are established. If an ${\cal R}$-compatible monoid variety contains a nonabelian group, then this variety is periodic and consists of completely regular monoids only.

AMS Subject Classification (1991): 20M07, 20M10

Received December 9, 2004, and in revised form August 16, 2005. (Registered under 5886/2009.)