In this paper we investigate the interplay between isolated suborders and closures. Isolated suborders are a special kind of suborders and can be used to diminish the number of elements of an ordered set by means of a quotient construction. The decisive point is that there are simple formulae establishing relationships between the number of closures in the original ordered set and the quotient thereof induced by isolated suborders. We show how these connections can be used to derive a recursive algorithm for counting closures, provided the ordered set under consideration contains suitable isolated suborders.

翻译：本文研究孤立子序与闭包之间的相互影响。孤立子序是一种特殊类型的子序，可通过商构造的方法减少偏序集中的元素数量。关键在于存在简洁的公式，能够建立原始偏序集中闭包数量与其由孤立子序诱导的商偏序集中闭包数量之间的关系。我们展示了如何利用这些关联推导出一种递归算法来计数闭包，前提是所考虑的偏序集包含合适的孤立子序。