  In this paper independent sets of closure operations are introduced. We
characterize minimal keys and antikeys of closure operations in terms of
independent sets. We establish an expression on the connection between minimal
keys and antikeys of closure operations based on independent sets. We construct
two combinatorial algorithms for finding all minimal keys and all antikeys of a
given closure operation based on independent sets. We estimate the time
complexity of these algorithms. Finally, we give an NP-complete problem
concerning nonkeys of closure operations.
