Injective envelope of graphs and transition systems

Injective envelopes of transition systems and Ferrers languages

“…A metric point of view for discrete structures, inspired from the work of Quilliot (1983), was applied first to posets and graphs, by the second author [37] and developped throught the theses of Jawhari (1983), Misane (1984) and [23] (1986). It was then extended to transition systems, [38] (1994), [42] (1992), [29] (1998), [30] (2018). In the case of transition systems, this point of view consists to consider arbitrary transition systems as metric spaces.…”

“…It was tempting to extend these results to the category of transitions systems and to the category of metric spaces over (A * ). This was partially done in [38] and developped in [28][29][30]42] as a by-product of the theory of metric spaces over a Heyting algebra developped in [23] and [38].…”

“…The study of such injective envelope was initiated in [29] and applied to the study of retracts of directed graphs. It was based on the properties of the injective envelope of two-element metric spaces.…”

A fixed point theorem for commuting families of relational homomorphisms. Applications to metric spaces, ordered sets and oriented graphs

“…n}, L u ) with end-points 0 and n (where n is the length ℓ(u) of u) such that This notion is due to Quilliot [34,35] (Quilliot considered reflexive directed graphs, not necessarily oriented, and in defining the distance, considered only oriented paths). A general study is presented in [17]; some developments appear in [37] and [21].…”

Graphs Admitting $k$-NU Operations. Part 1: The Reflexive Case

“…The analog for bipartite, undirected graphs is due to Bandelt [2]. It should be noted that for general reflexive digraphs, the class of absolute retracts is strictly larger than that of retracts of products of paths (but see [20] for an analogous description. )…”