  The concept of a $\lambda$-lattice was introduced by V. Sn\'a\v sel in order
to generalize some lattice concepts for directed posets whose elements need not
have suprema or infima. We extend the concept of semimodularity from lattices
to $\lambda$-lattices and show connections to the lower covering condition and
its generalizations. We further show that, contrary to the case of lattices,
for $\lambda$-lattices semimodularity and the (weak) lower covering condition
are independent properties. However, under some additional conditions
semimodularity implies the (weak) lower covering condition. Examples of
corresponding $\lambda$-lattices are presented.

Rings and Algebras