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<h1>8.1 Introduction</h1><p>
<p>
A poset is just a partially ordered set. To display posets
reasonably in a generic way we need additional structure. So for XGAP a
poset comes in so called levels. At all times in the life of a graphic
poset there are only finitely many levels and they are totally ordered,
that is for two levels we can always say, which one is ``higher''. The
position within the graphic sheet reflects this ordering.
<p>
The levels are parametrized by ``level parameters'', which can be any
<font face="Gill Sans,Helvetica,Arial">GAP</font> object but must be unique within a graphic poset. A level is always
accessed by its level parameter and <strong>not</strong> by its number!
<p>
The vertices in each level are grouped into classes. For example for 
graphic subgroup lattices vertices in the same class correspond to conjugate
subgroups, vertices in the same level have the same size or index in the
whole group. The classes within each level are parametrized by ``class
parameters'', which can be any <font face="Gill Sans,Helvetica,Arial">GAP</font> object but must be unique within a
level. A class within a level is always accessed by its class parameter and 
<strong>not</strong> by its number!
<p>
The user must supply a <strong>partial order</strong> for all of his levels. The mechanism 
to achieve this is the operation <code>CompareLevels</code>, which compares two level
parameters. The current <strong>total order</strong> of the levels is always a refinement
of the partial order. The user can permute levels, if that does not
contradict the partial order defined by <code>CompareLevels</code>.
<p>
A vertex in the poset that is ``contained in'' another vertex in the poset
order (we speak of ``inclusion'' like in the case of subgroup lattices)
must always be in a level that is lower on the screen, because there 
is only a connecting line representing the inclusion. This is achieved by
the fact, that inclusions of vertices are communicated to XGAP just by
creating an ``edge'' between them. This means, that the vertex in the
``lower'' level lies in the vertex in the ``higher'' level. There must not
be edges between vertices in the same level!
<p>
The terminology ``vertices'' and ``edges'' comes from the fact, that a
graphic poset is just a special case of a graphic graph, where vertices can 
be placed anywhere in the sheet and edges have nothing to do with
inclusion. It is planned that also a graphic graph library is implemented
in XGAP but it is not yet operational. However everything which could be 
done not only for posets but at the same time for graphs is implemented
already within the poset package. This explains the usage of ``graph'' in
many places where you would otherwise expect ``poset''.
<p>
What you have to do to use the graphic poset package is create a graphic
poset (a special instance of a graphic sheet), create some levels and
perhaps classes within them. Then you can create vertices and edges, to
encode the ordering. Everything else is done by the library. See the next
section for details about the available operations.
<p>
Note that we chose a functional approach for certain decision
procedures. This means that for example if you create a vertex and do not
specify a position, an operation (<code>ChoosePosition</code>) is called to determine
the actual position. You can use the generic routines or install your own
methods for all of those decisions. In this case you just set a new filter
for your posets and overload the generic methods by special routines for
objects with your new filter set. You can see this approach in the example
in <a href="C008S003.htm">An Example</a>.
<p>
<p>
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<address>xgap manual<br>Mai 2003
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