Local finiteness for Green relations in (I-)semigroup varieties

In this work, the lattice of varieties of semigroups and the lattice of varieties of I-semigroups (a common setting for both the variety of completely regular semigroups and the variety of inverse semigroups) are studied with respect to the following concepts: a variety V of (I-)semigroups is said to be locally K-finite, where K stands for any of the five Green's relations, if every finitely generated semigroup from V has only finitely many (distinct) K-classes.