Semimodular $\lambda$-lattices

Abstract

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.