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：本文研究了孤立子序与闭包之间的相互作用。孤立子序是一种特殊的子序，可用于通过商构造减少有序集的元素数量。关键在于存在简洁的公式，建立了原始有序集中闭包数量与其由孤立子序诱导的商集闭包数量之间的关系。我们展示了如何利用这些联系推导出计数闭包的递归算法，前提是所考虑的有序集包含合适的孤立子序。