Halfordered sets, halfordered chain structures and splittings by chains

摘要

Let $ (mathcal{P},frak{G}_{1}, frak{G}_{2}) $ be a 2-net where the set frak{C} of chains is not empty and let K s be a splitting of $mathcal{P}$ by a chain $ Kinfrak{C} $ . Then there correspond two halforders $xi_{l},xi_{r}$ of the set K which are related, and vice versa, if there are given two related halforders of K then there exists a splitting of $ mathcal{P} $ by K. The questions "when is $ xi_{l}=xi_{r} $ ?", "when is $ xi_{l} $ convex or an order?" will be studied. Moreover it will be shown that $ (mathcal{P},frak{G}_{1}, frak{G}_{2},{K_{s}}) $ can be embedded in a halfordered chain structure in the sense of [1].