Halfordered sets, halfordered chain structures and splittings by chains

Abstract.

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 $K\in\frak{C}$. 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].