Some Characterizations of Upper Semimodular Lattices and Posets

In this paper, we introduce two new conditions that give a distinctive description of upper semimodularity in finite lattices. Their relations with the existing conditions due to Mac Lane are established. Also, all these conditions for lattices are generalized for posets and relations amongst them are studied. We also give a forbidden structure as a characterization for finite upper semimodular posets. Several counterexamples are constructed.