Separation properties of 3-Steiner and 3-monophonic convexity in graphs

Let V be a finite set and C a collection of subsets of V. The ordered pair (V,C) is an alignment if C is closed under taking intersections and contains both 0̸ and V. If (V,C) is an alignment, then C is a convexity for V, and the elements of C are referred to as the convex sets of the convexity C. A convex set A is a half-space if V−A is convex. The following separation properties have been defined for a given convexity C of V.