  The basic distinction between already known algorithmic characterizations of
matroids and antimatroids is in the fact that for antimatroids the ordering of
elements is of great importance.
  While antimatroids can also be characterized as set systems, the question
whether there is an algorithmic description of antimatroids in terms of sets
and set functions was open for some period of time.
  This article provides a selective look at classical material on algorithmic
characterization of antimatroids, i.e., the ordered version, and a new
unordered version. Moreover we empathize formally the correspondence between
these two versions.
