闭包系统最大扩张的表示
Representations for the largest Extension of a closure system
摘要：
我们考虑有限集合S上的闭包系统的扩张作为包含给定闭包系统作为子格的同一集合S上的闭包系统。闭包系统可以用不同的方式表示，例如用蕴涵基或用它的交不可约元素的集合。当一个闭包系统是由一个蕴涵基描述时，我们提供了一个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一个小的修改的输入闭包系统的蕴涵基地。这回答了[12]中提出的一个问题。第二，我们感兴趣的是计算最大的扩张时，封闭系统是由它的所有满足不可约元素的集合。给出了求闭包系统最大扩张的增量式多项式时间算法，并在交不可约元素数呈指数增长时给出了左开闭包系统。
英文摘要：
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.
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