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.

翻译：考虑到隐含的基点,关闭系统的著名代表,与一定的一组的不一致的二进制关系,我们感兴趣的是将所有最大一致的封闭装置(由MCCEnnum简短地指出)都列出来的问题。我们显示,除非$\ textsf{P}=\textsf{NP}=\ textsf{NP}$,即使对于受约束程度较低的拉特克来说也是如此,否则中链Enum无法在输出-多极性状态下解决。我们给出了一个递增-多极性算法,用恒定的 Carath/'eodology数字解决封闭系统的中链Enum问题。我们最后证明,在双亚原子的原子封闭系统中,如果最小的发电机遵守一个独立状态(这种状态是非原子式模块拉特克特),中链门的关闭系统以前是由一个多元延迟算法解决的。