Mathematical semantics of the internal structure of algorithms

作为这项研究的结果，我们从集合层的普通定义中引入了完全偏序集层的新概念，它是通过用完全偏序集和连续函数的范畴CPO代替集合和函数的范畴集而给出的。这样得到的框架允许我们得到笛卡尔闭范畴CPOS(X)，在这个范畴中，我们可以为程序设计语言的指称语义开发各种结构，不包括可拓展性条件。这是因为为了解释编程语言的灵活语法装置，CPOS(X)具有许多所需的特性。

As a result of this research, we introduced a new notion of sheaves of complete partially ordered sets from the ordinary definition of sheavs of sets, which is given by replacing the category Sets of sets and functions with the category Cpos of complete partially ordered sets and continuous functions. The framework so obtained allows us to have the cartesian closed category Cpos(X) in which we can develop various structures for denotational semantics of programming languages excluding the condition of extensionality. This is because of a number of desirable features of Cpos(X) in order to interpret flexible syntactical devices of programming languages.