Antimatroids, Betweenness, Convexity

Korte and Lovász [12, 13] founded the theory of greedoids . These combinatorial structures characterize a class of optimization problems that can be solved by greedy algorithms. In particular, greedoids generalize matroids , introduced earlier by Whitney [16]. Antimatroids , introduced by Dilworth [3] as particular examples of semimodular lattices, make up another class of greedoids. Antimatroids are related to abstract convexity; let us explain how. Kay and Womble [11] defined a convexity space on a ground set E as a tuple (E,N ), where N is a collection of subsets of E such that ∅ ∈ N , E ∈ N , and N is closed under intersections. Members of N are called convex sets . The convex hull of a subset X of E is defined as the intersection of all convex supersets of X and is denoted by τN (X). Independently of each other, Edelman [6] and Jamison [9] initiated the study of convexity spaces (E,N ) with the anti-exchange property