Some characterizations of upper semimodular lattices and posets.
In the context of semimodularity in finite lattices, the authors have introduced two new conditions to describe upper semimodularity. The relationships between the new conditions and other classical ones due to Mac Lane are analyzed in depth. Then the authors furnish a generalization for posets of all these conditions previously introduced for semimodular lattices. In this direction, in order to characterize finite upper semimodular posets, the authors have discovered a forbidden structure which is studied in a section of the manuscript. Concerning all the items introduced and considered, the authors also give suitable counterexamples. A final table of implications showing the hierarchies among the concepts and conditions introduced and analyzed closes the paper.
scientific article; zbMATH DE number 6568272