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，可通过商结构构造来减少有序集中的元素数量。关键之处在于，存在简单的公式能够建立原始有序集中闭包数量与其由孤立子序诱导的商结构之间的关联。我们展示了如何利用这些联系推导出一种递归算法以计数闭包，前提是所考虑的有序集包含适当的孤立子序。