Rank axiom of modular supermatroids: A connection with directional DR submodular functions

A matroid has been one of the most important combinatorial structures since it was introduced by Whitney as an abstraction linear independence. As property a matroid, can be characterized several different (but equivalent) axioms, such augmentation, base exchange, or rank axiom. supermatroid is generalization defined on lattices. Here, central question whether equivalent axioms similar to matroid. Barnabei, Nicoletti, and Pezzoli supermatroids distributive lattices, Fujishige, Koshevoy, Sano generalized results for cg-matroids (supermatroids lower locally lattices). In this study, we focus modular which are superclass provide characterizations We characterize lattices using axiom in function directional DR-submodular function, submodular authors. Using characterization based functions, further prove strong exchange supermatroid, application optimization. also reveal relation between semimodular common lattice lattice.