Extending a partially ordered set: Links with its lattice of ideals

A well-known result of Bonnet and Pouzet [3] bijectively links the set of linear extensions of a partial order P with the set of maximal chains of its lattice of ideals I(P). We extend this result by showing that there is a one-to-one correspondence between the set of all extensions of P and the set of all sublattices of I(P) which are chain-maximal in the sense that every chain which is maximal (for inclusion) in the sublattice is also maximal in the lattice. We prove that the absence of an order S as a convex suborder of P is equivalent to the absence of I(S) as a convex suborder of I(P). Let S be a set of partial orders and let us call S-convex-free any order that does not contain any order of S as convex suborder. We deduce from the previous results that there is a one-to-one correspondence between the set of S-convex-free extensions of P and the set of I(S)-convex-free chain-maximal sublattices of I(P). This can be applied to some classical classes of orders (total orders and in the finite case, weak orders, interval orders, N-free orders). In the particular case of total orders this gives as a corollary the result of Bonnet and Pouzet.