%0 Journal Article %T Interval Reductions and Extensions of Orders : Bijections to Chains in Lattices %+ Freie Universität Berlin %+ Software Tools for Telecommunications and Distributed Systems (RESEDAS) %A Felsner, Stefan %A Gustedt, Jens %A Morvan, Michel %Z Article dans revue scientifique avec comité de lecture. %< avec comité de lecture %Z 99-R-283 || felsner99a %@ 0167-8094 %J Order %I Springer Verlag %V 15 %N 3 %P 221-246 %8 1999 %D 1999 %K weak order %K bijection %K chains %K interval extension %K interval reduction %K latticeof antichains %K chaînes %K extensions intervallaires %K reductions intervallaires %K treillis des antichaînes %K ordres forts %Z Computer Science [cs]/Other [cs.OH]Journal articles %X We discuss bijections that relate families of chains in lattices associated to an order $P$ and families of interval orders defined on the ground set of $P$. Two bijections of this type have been known: (1) The bijection between maximal chains in the antichain lattice $AA(P)$ and the linear extensions of $P$. (2) A bijection between maximal chains in the lattice of maximal antichains $AM(P)$ and minimal interval extensions of $P$. We discuss two approaches to associate interval orders to chains in $AA(P)$. This leads to new bijections generalizing Bijections~1 and~2. As a consequence we characterize the chains corresponding to weak-order extensions and minimal weak-order extensions of $P$. Seeking for a way of representing interval reductions of $P$ by chains we came up with the separation lattice $SL(P)$. Chains in this lattice encode an interesting subclass of interval reductions of $P$. Let $SLM(P)$ be the lattice of maximal separations in the separation lattice. Restricted to maximal separations the above bijection specializes to a bijection which nicely complements 1 and 2. (3) A bijection between maximal chains in the lattice of maximal separations $\SLM(P)$ and minimal interval reductions of $P$. %G English %L inria-00098826 %U https://inria.hal.science/inria-00098826 %~ CNRS %~ INRIA %~ INPL %~ IRISA %~ INRIA-LORRAINE %~ LORIA2 %~ INRIA-NANCY-GRAND-EST %~ TESTALAIN1 %~ UNIV-LORRAINE %~ INRIA2 %~ LORIA %~ UR1-MATH-STIC %~ UR1-UFR-ISTIC %~ INRIA-300009 %~ UR1-MATH-NUM %~ INRIA-ALLEMAGNE