Seeking the largest solution to an expression of the form A x <= B is a common task in several domains of engineering and computer science. This largest solution is commonly called quotient. Across domains, the meanings of the binary operation and the preorder are quite different, yet the syntax for computing the largest solution is remarkably similar. This paper is about finding a common framework to reason about quotients. We only assume we operate on a preorder endowed with an abstract monotonic multiplication and an involution. We provide a condition, called admissibility, which guarantees the existence of the quotient, and which yields its closed form. We call preordered heaps those structures satisfying the admissibility condition. We show that many existing theories in computer science are preordered heaps, and we are thus able to derive a quotient for them, subsuming existing solutions when available in the literature. We introduce the concept of sieved heaps to deal with structures which are given over multiple domains of definition. We show that sieved heaps also have well-defined quotients.

中文翻译：

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先序理论中的商

寻求 A x <= B 形式表达式的最大解是工程和计算机科学多个领域的常见任务。这个最大的解通常称为商。跨域，二元运算和前序的含义大不相同，但计算最大解的语法却非常相似。这篇论文是关于寻找一个通用的框架来推理商。我们只假设我们对一个具有抽象单调乘法和对合的前序进行操作。我们提供了一个条件，称为可容许性，它保证商的存在，并产生它的封闭形式。我们将满足可接纳性条件的结构称为预序堆。我们证明了计算机科学中的许多现有理论都是预先排序的堆，因此，我们能够为它们推导出商数，包括文献中可用的现有解决方案。我们引入了筛选堆的概念来处理在多个定义域上给出的结构。我们表明筛分堆也有明确定义的商。