Modularity of the lattice
of normal subgroups of a group is well-known. Equivalently, the lattice of
congruence relations on a group is a modular lattice. A natural question
to consider is how far can we push the last statement when dealing with
the larger class semigroups. It is easily shown that the class of congruence
lattices of semigroups satisfies no nontrivial lattice identity. Thus we might
try to find those semigroups whose congruence lattice is a modular lattice.
This problem is of all the more interest due to the fact that congruences on
algebras whose congruence lattice is a modular lattice satisfy variants of the
Jordan-Holder-Schreier theorem. In this paper we show that the commutative
cancellative semigroups whose congruence lattice is a modular lattice are the abelian
groups, the positive cones of rational groups, and the nonnegative cones of
rational groups. We also show that the commutative cancellative semigroups
with a distributive lattice of congruences are locally cyclic or locally cyclic
with an identity adjoined. This last result generalizes Ore’s theorem that a
group has a distributive lattice of congruences if and only if it is locally
cyclic.
