2020/02/07 by Mustapha Kabil, Kabil, Mustapha, Maurice Pouzet +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Advanced Topology and Set Theory #Amino Acid Enzymes and Metabolism #Combinatorics (math.CO) #FOS: Mathematics #Functional Equations Stability Results #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2002.03019
openalex publication_date 2020/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this survey we present a generalization of the notion of metric space and\nsome applications to discrete structures as graphs, ordered sets and transition\nsystems. Results in that direction started in the middle eighties based on the\nimpulse given by Quilliot (1983). Graphs and ordered sets were considered as\nkind of metric spaces, where - instead of real numbers - the values of the\ndistance functions d belong to an ordered semigroup equipped with an\ninvolution. In this frame, maps preserving graphs or posets are exactly the\nnonexpansive mappings (that is the maps f such that d(f(x),f(y))\≤\nd(x,y), for all x,y). It was observed that many known results on retractions\nand fixed point property for classical metric spaces (whose morphisms are the\nnonexpansive mappings) are also valid for these spaces. For example, the\ncharacterization of absolute retracts, by Aronszajn and Panitchpakdi (1956),\nthe construction of the injective envelope by Isbell (1965) and the fixed point\ntheorem of Sine and Soardi (1979) translate into the Banaschewski-Bruns theorem\n(1967), the MacNeille completion of a poset (1933) and the famous Tarski fixed\npoint theorem (1955). This prompted an analysis of several classes of discrete\nstructures from a metric point of view. In this paper, we report the results\nobtained over the years with a particular emphasis on the fixed point property.\n