  A function $F$ defined on all subsets of a finite ground set $E$ is
quasi-concave if $F(X\cup Y)\geq\min\{F(X),F(Y)\}$ for all $X,Y\subset E$.
Quasi-concave functions arise in many fields of mathematics and computer
science such as social choice, theory of graph, data mining, clustering and
other fields.
  The maximization of quasi-concave function takes, in general, exponential
time. However, if a quasi-concave function is defined by associated monotone
linkage function then it can be optimized by the greedy type algorithm in a
polynomial time.
  Quasi-concave functions defined as minimum values of monotone linkage
functions were considered on antimatroids, where the correspondence between
quasi-concave and bottleneck functions was shown (Kempner & Levit, 2003). The
goal of this paper is to analyze quasi-concave functions on different families
of sets and to investigate their relationships with monotone linkage functions.
