Jean-Xavier Rampon

What is reconstruction for ordered sets

Reconstruction questions arise when studying interactions between the isomorphic type of a structure and the isomorphic types of its substructures. In this survey paper we are interested in binary relations and particularly we focus on partially ordered binary relations. We present most of the known results on partially ordered sets and that for different kinds of reconstruction: among them we have the [emailxa0protected]?sse-reconstruction, the Ulam-reconstruction, the max-reconstruction and the set-reconstruction.

Reconstruction of Posets with the Same Comparability Graph

Given two finite posetsPandP? with the same comparability graph, we show that if |V(P)|?4 and if for allx?V(P),P?x?P??x, thenP?P?. This result leads us to characterize the finite posetsPsuch that for allx?V(P),P?x?P*?x.

Partial orders and their convex subsets

Abstract We introduce some visibility relations between convex subsets of a partial order that are partial orders themselves. As a consequence we obtain a general framework for partial orders providing an interesting coding, and some new characterizations of some known classes of partial orders.

Reconstruction of finite truncated semi-modular lattices

In this paper, we prove reconstruction results for truncated lattices. The main results are that truncated lattices that contain a 4-crown and truncated semi-modular lattices are reconstructible. Reconstruction of the truncated lattices not covered by this work appears challenging. Indeed, the remaining truncated lattices possess very little lattice-typical structure. This seems to indicate that further progress on the reconstruction of truncated lattices is closely correlated with progress on reconstructing ordered sets in general.

Partial Orders on Weak Orders Convex Subsets

We study a visibility relation on the nonempty connected convex subsets of a finite partially ordered set and we investigate the partial orders representable as a visibility relation of such subsets of a weak order. Moreover, we consider restrictions where the subsets of the weak order are total orders or isomorphic total orders.

Inductive Characterizations of Finite Interval Orders and Semiorders

We introduce an inductive definition for two classes of orders. By simple proofs, we show that one corresponds to the interval orders class and that the other is exactly the semiorders class.

Chain Dominated Orders

We study finite partial orders which have a chain such that every element of the order either belongs to this chain or has all its covers in this chain. We show that such orders are exactly the orders being both interval orders and truncated lattices. We prove that their jump number is polynomially tractable and that their dimension is unbounded. We also show that every order admits a visibility model having such an order as host.