  Given an implicational base, a well-known representation for a closure
system, an inconsistency binary relation over a finite set, we are interested
in the problem of enumerating all maximal consistent closed sets (denoted by
MCCEnum for short). We show that MCCEnum cannot be solved in output-polynomial
time unless $\textsf{P} = \textsf{NP}$, even for lower bounded lattices. We
give an incremental-polynomial time algorithm to solve MCCEnum for closure
systems with constant Carath\'eodory number. Finally we prove that in biatomic
atomistic closure systems MCCEnum can be solved in output-quasipolynomial time
if minimal generators obey an independence condition, which holds in atomistic
modular lattices. For closure systems closed under union (i.e., distributive),
MCCEnum has been previously solved by a polynomial delay algorithm.
