2.81 (1966)

a) Does there exist an axiomatizable class of lattices $\mathfrak{K}$ such that the lattice of all subsemigroups of a semigroup $S$ is isomorphic to some lattice in $\mathfrak{K}$ if and only if $S$ is a free group?
b) The same question for free abelian groups.

Analogous questions have affirmative answers for torsion-free groups, for non-periodic groups, for abelian torsion-free groups, for abelian non-periodic groups, for orderable groups (the corresponding classes of lattices are even finitely axiomatizable). Thus, in posed questions one may assume from the outset that the semigroup $S$ is a torsion-free group (respectively, a torsion-free abelian group).