Editing Greedoid (section)

==Classes==
Most classes of greedoids have many equivalent definitions in terms of set system, language, poset, [[simplicial complex]], and so on. The following description takes the traditional route of listing only a couple of the more well-known characterizations.

An '''interval greedoid''' {{math|(''F'', ''E'')}} is a greedoid that satisfies the ''Interval Property'':

* if <math>A,B,C \in F</math> with <math>A \subseteq B \subseteq C,</math> then, for all <math>x \in E \setminus C:</math>  
<math display=block>
\begin{matrix} A \cup \{x\} \in F \\ 
C \cup \{x\} \in F \end{matrix}
\implies B \cup \{x\} \in F.</math>

Equivalently, an interval greedoid is a greedoid such that the union of any two feasible sets is feasible if it is contained in another feasible set.

An '''[[antimatroid]]''' {{math|(''F'', ''E'')}} is a greedoid that satisfies the ''Interval Property without Upper Bounds'':

* if {{tmath|A,B \in F}} with {{tmath|A \subseteq B,}} then, for all {{tmath|x \in E \setminus B,}} {{tmath|A \cup \{x\} \in F}} implies {{tmath|B \cup \{x\} \in F.}}

Equivalently, an antimatroid is (i) a greedoid with a unique basis; or (ii) an accessible set system closed under union.  It is easy to see that an antimatroid is also an interval greedoid.

A '''[[matroid]]''' {{math|(''F'', ''E'')}} is a non-empty greedoid that satisfies the ''Interval Property without Lower Bounds'':

* if {{tmath|B,C \in F}} with {{tmath|B \subseteq C,}} then, for all {{tmath|x \in E \setminus C,}} {{tmath|C \cup \{x\} \in F}} implies {{tmath|B \cup \{x\} \in F.}}

It is easy to see that a matroid is also an interval greedoid.