Halfordered sets, halfordered chain structures and splittings by chains (Q1396028)

The authors continue the study of halforders and chains started in [\textit{B. Alinovi}, \textit{H. Karzel} and \textit{C. Tonesi}, Quaderni del Sem. Mat. d. Brescia 18 (1999)] and continued in [J. Geom. 71, No. 1--2, 1--18 (2001; Zbl 1013.51002)]. Here they consider a \(2\)-net \(({\mathcal P},{\mathcal S}_1,{\mathcal S}_2)\) whose set of chains is nonempty. Then to each splitting \(K_s\) of \({\mathcal P}\) by a chain \(K\) there correspond two halforders \(\xi_l\), \(\xi_r\) of \(K\) that are related; that is for all \(a,b,c,d\in K\) there holds \([a,b| c,d]\xi_l=[c,d| a,b]\xi_r\). Conversely, given two related halforders \(\xi_l\), \(\xi_r\) of the chain \(K\) there is a splitting of \({\mathcal P}\) by \(K\). The authors study the effect of various conditions imposed on the halforders \(\xi_l\), \(\xi_r\) (for example, the conditions \(\xi_l= \xi_r\) or \(\xi_l\) is an order). More particularly, they show that \(({\mathcal P},{\mathcal S}_1,{\mathcal S}_2,\{K_s\})\) can be imbedded in a halfordered chain structure. For details and definitions the interested reader is referred to the article and its predecessors.