Semimodular $\lambda$-lattices

TL;DR

This paper extends the concept of semimodularity to $mbda$-lattices, exploring their properties, relationships with covering conditions, and providing examples, thus broadening the understanding of lattice generalizations.

Contribution

It introduces and analyzes semimodularity in $mbda$-lattices, establishing new connections and independence results compared to classical lattice theory.

Findings

Semimodularity and lower covering condition are independent in $mbda$-lattices.

Under certain conditions, semimodularity implies the lower covering condition.

Examples of $mbda$-lattices illustrating these properties are provided.

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.