Interval Reductions and Extensions of Orders : Bijections to Chains in Lattices
Résumé
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 $P$.