Interval reductions and extensions of orders: Bijections to chains in lattices

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 A(P) and the linear extensions of P. (2) The bijection between maximal chains in the lattice of maximal antichains A(M)(P) and minimal interval extensions of P. We discuss two approaches to associate interval orders with chains in A(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 upon the separation lattice S(P). Chains in this lattice encode an interesting subclass of interval reductions of P. Let S-M(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 S-M(P) and minimal interval reductions of P.