Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set (Q2826343)

The authors study sublattices of the lattice \(\mathcal {C}(X,\leq)\) of all convex subsets of a linearly ordered set \(X\) by means of ternary betweenness relations induced by certain families \(\mathcal M\) of subsets of \(X\). In their main result, they formulate necessary and sufficient conditions for a family \(\mathcal M\) of subsets of \(X\) completely separating \(X\) to be a sublattice of \(\mathcal {C}(X,\leq)\) with respect to some linear order \(\leq \) on \(X\). As an application they characterize the following two types of Hausdorff topological spaces: (1) spaces with a base formed by the open intervals of some linear order on \(X\) (orderable spaces), (2) spaces having a base whose all members are convex subsets of \(X\) with respect to some linear order on \(X\) (suborderable spaces).