C-independence and c-rank of posets and lattices

Continuing with the authors concept (and results) of defining independence for columns of a boolean and superboolean matrix, we apply this theory to finite lattices and finite posets, introducing boolean and superboolean matrix representations for these objects. These representations yield the new concept of c-independent subsets of lattices and posets, for which the notion of c-rank is determined as the cardinality of the largest c-independent subset. We characterize this c-rank and show that c-independent subsets have a very natural interpretation in term of the maximal chains of the Hasse diagram and the associated partitions of the lattice. This realization has direct important connections with chamber systems.