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Interval Reductions and Extensions of Orders: Bijections to Chains in Lattices

Felsner, Stefan; Gustedt, Jens; Morvan, Michel

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

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:par (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 AAM(P) and minimal interval extensions of P. par 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. par Seeking for a way of representing interval reductions of P by chains we came up with the separation lattice S(P). Chains in this lattice encode an interesting subclass of interval reductions of P. Let SM(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.par (3) A bijection between maximal chains in the lattice of maximal separations SM(P) and minimal interval reductions of P.