Directed complete poset congruences
Directed complete poset congruences
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DOI：
10.1016/j.jpaa.2019.01.003
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发表时间：
2019-10
影响因子：
0.8
通讯作者：
M. Mahmoudi;Halimeh Moghbeli;K. Pióro
中科院分区：
文献类型：
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作者：
M. Mahmoudi;Halimeh Moghbeli;K. Pióro
对有序结构的代数性质的研究表明，它们在许多情况下的行为与代数结构不同。例如，将满射映射刻画为模其核关系的商集的集合的基本映射定理的类似物，对于偏序集（偏序集）之间的保序映射是不正确的。本文的主要目的是研究有向完全偏序集（dcpo）的逆映射及其与满射dcpo映射（有向连接保持映射）的关系。研究这类无穷序结构的动机是它们在Domain理论中的重要性，Domain理论是一个介于数学和理论计算机科学之间的边缘理论.本文引入dcpo（directed complete partially ordered sets）上的预同余的概念，给出了dcpo同余的一个刻画.另外，还证明了与自然dcpo同余不同，dcpo同余是满射dcpo映射的核。此外，虽然它是已知的，图像的dcpo地图不一定是subdcpo的余域，我们发现等价条件的dcpo地图，以满足这一性质。此外，我们还证明了dcpo映射的分解定理及其推论。
The study of algebraic properties of ordered structures has shown that their behavior in many cases is different from algebraic structures. For example, the analogues of the fundamental mapping theorem for sets which characterizes surjective maps as quotient sets modulo their kernel relations, is not true for order-preserving maps between posets (partially ordered sets). The main objective of this paper is to study the quotients of dcpos (directed complete partially ordered sets), and their relations with surjective dcpo maps (directed join preserving maps). The motivation of studying such infinitary ordered structures is their importance in domain theory, a theory on the borderline of mathematics and theoretical computer science.In this paper, introducing the notion of a pre-congruence on dcpos (directed complete partially ordered sets), we give a characterization of dcpo congruences. Also, it is proved that unlike natural dcpo congruences, the dcpo congruences are precisely kernels of surjective dcpo maps. Also, while it is known that the image of a dcpo map is not necessarily a subdcpo of its codomain, we find equivalent conditions on a dcpo map to satisfy this property. Moreover, we prove the Decomposition Theorem and its consequences for dcpo maps.