Limits of Structures and the Example of Tree-Semilattices

The notion of left convergent sequences of graphs introduced by Lov\' asz et al. (in relation with homomorphism densities for fixed patterns and Szemer\'edi's regularity lemma) got increasingly studied over the past $10$ years. Recently, Ne\v set\v ril and Ossona de Mendez introduced a general framework for convergence of sequences of structures. In particular, the authors introduced the notion of $QF$-convergence, which is a natural generalization of left-convergence. In this paper, we initiate study of $QF$-convergence for structures with functional symbols by focusing on the particular case of tree semi-lattices. We fully characterize the limit objects and give an application to the study of left convergence of $m$-partite cographs, a generalization of cographs

Conservative median algebras and semilattices

We characterize conservative median algebras and semilattices by means of forbidden substructures and by providing their representation as chains. Moreover, using a dual equivalence between median algebras and certain topological structures, we obtain descriptions of the median-preserving mappings between products of finitely many chains

Infinite combinatorial issues raised by lifting problems in universal algebra

The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results thus encountered. The corresponding problematic is articulated around the notion of a k-ladder (for proving that a critical point is large), large free set theorems and the classical notation (k,r,l){\to}m (for proving that a critical point is small). In the middle, we find l-lifters of posets and the relation (k, < l){\to}P, for infinite cardinals k and l and a poset P.Comment: 22 pages. Order, to appea

Byzantine Approximate Agreement on Graphs

Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that 1) the output values are in the convex hull of the non-faulty processors\u27 input values, 2) the output values are within distance d of each other. Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures

Trimness of Closed Intervals in Cambrian Semilattices

In this article, we give a short algebraic proof that all closed intervals in a $\gamma$-Cambrian semilattice $\mathcal{C}_{\gamma}$ are trim for any Coxeter group $W$ and any Coxeter element $\gamma\in W$. This means that if such an interval has length $k$, then there exists a maximal chain of length $k$ consisting of left-modular elements, and there are precisely $k$ join- and $k$ meet-irreducible elements in this interval. Consequently every graded interval in $\mathcal{C}_{\gamma}$ is distributive. This problem was open for any Coxeter group that is not a Weyl group.Comment: Final version. The contents of this paper were formerly part of my now withdrawn submission arXiv:1312.4449. 12 pages, 3 figure