Rencontre à Marseille, 17-19 novembre 2010

Résumé : L'ensemble des mots sur un alphabet donné, tels que chaque lettre a une nombre fixé d'occurrences, est un treillis qui généralise l'ordre faible de Bruhat sur les permutations. On l'appelle treillis des multi-permutations ou treillis de chemins. Si l'on fixe seulement un alphabet de taille n et on laisse varier le nombre d'occurrences des lettres, on obtient une famille inductive de treillis. La colimite de cette famille (c'est à dire, la réunion) est un treillis qui contient tous les mots. Quand on considère un mot comme un chemin rationnel dans le cube à n dimensions, ce treillis contient tous les chemins rationnels. Nous allons donner une description précise de la complétion de Dedekind-MacNeille de ce treillis. Nous montrerons que tout chemin continu dans le cube à n dimensions peut se coder comme un élément de ce treillis.

Résumé : We study the structure of existence varieties of complemented modular lattices and regular rings. In particular, we prove an HSP-type theorem and also consider the decidability problem for equational theories of those.

Résumé : Cyclic ordering relations allow to express betweenness relations, e. g. in periodic processes with concurrency. Two important differences with respect to acyclic ordering relations arise: the need for ternary models - since binary relations do not allow to distinguish rotational directions - , and the fact that sequential extensions, or orientations, do not always exist. This orientation problem for ternary cyclic order relations has been attacked in the literature from combinatorial perspectives, through rotation operations, and by connection with Petri nets; here, we propose a two-fold characterization of orientable cyclic orders in terms of symmetries of partial orders as well as in terms of separating sets (cuts). The results are inspired by properties of non-sequential properties; however, they also apply to dense structures, without restrictions on the cardinality.

Résumé : An operation on a set A is a map from An to A, where n is the arity of the operation. An algebra is a nonempty set endowed with a collection of operations. A congruence of an algebra A is an equivalent relation compatible with the operations. The lattice of congruences of A, dentoed Con A is the set of all congruences of A. It is a lattice for inclusion. The set of all finitely generated congruences of A is a (∨, 0)-semilattice, denoted by Conc A. If a (∨, 0)-semilattice S is isomorphic to Conc A, we say that A is a lifting of S. Let V be a variety (equational class) of algebras. The congruence class of V, denoted by Conc V is the class of all (∨, 0)-semilattice with a lifting in V. The congruence class of very few varieties of algebras have a good description. There is no chance in general to find such description. However, often it seems possible to describe inclusion of classes of congruences. Let V and W be varieties of algebras. The critical point between V and W is defined as

crit(V; W) = min{card S | S ∈ (Conc V) − (Conc W)}, if Conc V ⊆ Conc W,
             ∞, if Conc V ⊆ Conc W.

In this talk, we show tools that are used to compute critical points and example of their uses.