论文标题
广义度量空间。与图形，有序集和自动机的关系：调查
Generalized metric spaces. Relations with graphs, ordered sets and automata : A survey
论文作者
论文摘要
在这项调查中，我们介绍了度量空间概念的概括，以及某些应用程序作为图形，有序集和过渡系统的离散结构的应用。结果始于八十年代中期，基于Quilliot（1983）给出的冲动。图和有序集被视为一种公制空间，其中 - 而不是实数 - 距离函数$ d $的值属于配备有限的有序半群。在此框架中，保留图形或posets的地图正是非专用映射（即$ d（f（x），f（y），f（y））\ leq d（x，y）$，对于所有$ x，y $）。据观察，经典度量空间的缩回和固定点特性（其形态为非专用映射）的许多已知结果也适用于这些空间。 For example, the characterization of absolute retracts, by Aronszajn and Panitchpakdi (1956), the construction of the injective envelope by Isbell (1965) and the fixed point theorem of Sine and Soardi (1979) translate into the Banaschewski-Bruns theorem (1967), the MacNeille completion of a poset (1933) and the famous Tarski fixed point theorem （1955）。从公制的角度来看，这促使对几类离散结构进行了分析。在本文中，我们报告了多年来获得的结果，并特别强调了固定点属性。
In this survey we present a generalization of the notion of metric space and some applications to discrete structures as graphs, ordered sets and transition systems. Results in that direction started in the middle eighties based on the impulse given by Quilliot (1983). Graphs and ordered sets were considered as kind of metric spaces, where - instead of real numbers - the values of the distance functions $d$ belong to an ordered semigroup equipped with an involution. In this frame, maps preserving graphs or posets are exactly the nonexpansive mappings (that is the maps $f$ such that $d(f(x),f(y))\leq d(x,y)$, for all $x,y$). It was observed that many known results on retractions and fixed point property for classical metric spaces (whose morphisms are the nonexpansive mappings) are also valid for these spaces. For example, the characterization of absolute retracts, by Aronszajn and Panitchpakdi (1956), the construction of the injective envelope by Isbell (1965) and the fixed point theorem of Sine and Soardi (1979) translate into the Banaschewski-Bruns theorem (1967), the MacNeille completion of a poset (1933) and the famous Tarski fixed point theorem (1955). This prompted an analysis of several classes of discrete structures from a metric point of view. In this paper, we report the results obtained over the years with a particular emphasis on the fixed point property.
