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.

中文翻译：

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枚举封闭系统中的最大一致封闭集

给定一个隐含基础，一个闭包系统的著名表示形式，一个有限集上的不一致二进制关系，我们对枚举所有最大一致封闭集（简称为MCCEnum）的问题感兴趣。我们证明，除非$ \ textsf {P} = \ textsf {NP} $，即使对于下界晶格，也不能在输出多项式时间内求解MCCEnum。我们给出了一个递增多项式时间算法来求解具有恒定Carath'eodory数的封闭系统的MCCEnum。最后，我们证明了在双原子原子闭合系统中，如果最小生成器遵循一个独立条件，则MCCEnum可以在输出拟多项式时间内求解，该条件成立于原子模块化晶格中。对于在联合（即分配）下关闭的封闭系统，MCCEnum先前已通过多项式延迟算法求解。