Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..

Representation of ideals of relational structures

11International audienceThe \textit{age} of a relational structure $\mathfrak A$ of signature $\mu$ is the set $age(\mathfrak A)$ of its finite induced substructures, considered up to isomorphism. This is an ideal in the poset $\Omega_\mu$ consisting of finite structures of signature $\mu$ and ordered by embeddability. We shall show, that if the structures have infinitely many relations and if, among those, infinitely many are at least binary then there are ideals which do not come from an age. We provide many examples. We particularly look at metric spaces and offer several problems. We also answer a question due to R.~ Cusin and J.F. ~Pabion (1970): there is an ideal $I$ of isomorphism types of at most countable structures whose signature consists of a single ternary relation symbol such that $I$ does not come from the set _{\mathfrak I }(\mathfrak A)$of isomorphism types of substructures of$\mathfrak A$induced on the members of an ideal$\mathfrak I$ of sets

Representation of ideals of relational structures

Abstract The age of a relational structure A of signature µ is the set age(A) of its finite induced substructures, considered up to isomorphism. This is an ideal in the poset Ω µ consisting of finite structures of signature µ and ordered by embeddability. If the structures are made of infinitely many relations and if, among those, infinitely many are at least binary then there are ideals which do not come from an age. We provide many examples. We particularly look at metric spaces and offer several problems. We also provide an example of an ideal I of isomorphism types of at most countable structures whose signature consists of a single ternary relation symbol. This ideal does not come from the set age I (A) of isomorphism types of substructures of A induced on the members of an ideal I of sets. This answers a question due to R. Cusin and J.F. Pabion (1970)