Editing Matroid (section)

== Additional terms <span class="anchor" id="Additional_terms_anchor></span> == <!--Linked to by [[Simple matroid]] (redirect)-->
Let ''M'' be a matroid with an underlying set of elements ''E''.

* ''E'' may be called the ''ground set'' of ''M''.  Its elements may be called the ''points'' of ''M''.

* A subset of ''E'' ''spans'' ''M'' if its closure is ''E''.  A set is said to ''span'' a closed set ''K'' if its closure is ''K''.

* The [[matroid girth|girth]] of a matroid is the size of its smallest circuit or dependent set.

* An element that forms a single-element circuit of ''M'' is called a ''loop''. Equivalently, an element is a loop if it belongs to no basis.<ref name=Ox13/><ref name=Wh86130>{{harvp|White|1986|p=130}}</ref>

* An element that belongs to no circuit is called a ''coloop'' or ''isthmus''. Equivalently, an element is a coloop if it belongs to every basis.

: Loop and coloops are mutually dual.<ref name=Wh86130/>

* If a two-element set {''f, g''} is a circuit of ''M'', then ''f'' and ''g'' are ''parallel'' in ''M''.<ref name=Ox13/>

* A matroid is called ''simple'' if it has no circuits consisting of 1 or 2&nbsp;elements. That is, it has no loops and no parallel elements. The term ''combinatorial geometry'' is also used.<ref name=Ox13>{{harvp|Oxley|1992|p=13}}</ref>  A simple matroid obtained from another matroid ''M'' by deleting all loops and deleting one element from each 2-element circuit until no 2&nbsp;element circuits remain is called a ''simplification'' of ''M''.<ref name=Ox52>{{harvp|Oxley|1992|p=52}}</ref> A matroid is ''co-simple'' if its dual matroid is simple.<ref name=Ox347>{{harvp|Oxley|1992|p=347}}</ref>

* A union of circuits is sometimes called a ''cycle'' of ''M''. A cycle is therefore the complement of a flat of the dual matroid.  (This usage conflicts with the common meaning of "cycle" in graph theory.)

* A ''separator'' of ''M'' is a subset ''S'' of ''E'' such that <math>r(S) + r(E-S) = r(M)</math>. A ''proper'' or ''non-trivial separator'' is a separator that is neither ''E'' nor the empty set.<ref name=Ox128>{{harvp|Oxley|1992|p=128}}</ref> An ''irreducible separator'' is a non-empty separator that contains no other non-empty separator. The irreducible separators partition the ground set ''E''.

* A matroid that cannot be written as the direct sum of two nonempty matroids, or equivalently that has no proper separators, is called ''connected'' or ''irreducible''. A matroid is connected if and only if its dual is connected.<ref name=Wh86110>{{harvp|White|1986|p=110}}</ref>

* A maximal irreducible submatroid of ''M'' is called a ''component'' of ''M''. A component is the restriction of ''M'' to an irreducible separator, and contrariwise, the restriction of ''M'' to an irreducible separator is a component. A separator is a union of components.<ref name=Ox128/>

* A matroid ''M'' is called a ''frame matroid'' if it, or a matroid that contains it, has a basis such that all the points of ''M'' are contained in the lines that join pairs of basis elements.<ref>{{harvp|Zaslavsky|1994}}</ref>

* A matroid is called a [[paving matroid]] if all of its circuits have size at least equal to its rank.<ref name=Ox26>{{harvp|Oxley|1992|p=26}}</ref>

* The [[matroid polytope|basis polytope]] <math>P_M</math> is the [[convex hull]] of the [[indicator vector]]s of the bases of <math>M</math>

* The independence polytope of <math>M</math> is the [[convex hull]] of the [[indicator vector]]s of the independent sets of <math>M</math>.