Halfordered sets, halfordered chain structures and splittings by chains

摘要

Let (P,G_1,G_2) be a 2-net where the set C of chains is not empty and let K_s be a splitting of P by a chain K ∈ C. Then there correspond two halforders ξ_l,ξ_r of the set which are related, and vice versa, if there are given two related halforders of K then there exists a splitting of P by K. The questions "when is ξ_l = ξ_r?", "when is ξ_l convex or an order?" will be studied. Moreover it will be shown that (P,G_1,G_2,{K_s}) can be embedded in a halfordered chain structure in the sense of [1].