We consider extension of a closure system on a finite set S as a closure system on the same set S containing the given one as a sublattice. A closure system can be represented in different ways, e.g. by an implicational base or by the set of its meet-irreducible elements. When a closure system is described by an implicational base, we provide a characterization of the implicational base for the largest extension. We also show that the largest extension can be handled by a small modification of the implicational base of the input closure system. This answers a question asked in [12]. Second, we are interested in computing the largest extension when the closure system is given by the set of all its meet-irreducible elements. We give an incremental polynomial time algorithm to compute the largest extension of a closure system, and left open if the number of meet-irreducible elements grows exponentially.

中文翻译：

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封闭系统最大扩展的表示

我们将有限集 S 上的闭包系统的扩展视为同一集合 S 上的闭包系统，其中包含给定的子格。一个闭包系统可以用不同的方式表示，例如由一个蕴涵基或由它的集合不可约元素来表示。当闭包系统由蕴涵基描述时，我们提供最大扩展的蕴涵基的特征。我们还表明，最大的扩展可以通过对输入闭包系统的蕴涵基础进行小的修改来处理。这回答了 [12] 中提出的问题。其次，我们有兴趣计算当闭包系统由其所有相遇不可约元素的集合给出时的最大扩展。我们给出了一个增量多项式时间算法来计算闭包系统的最大扩展，