BigObject PartiallyOrderedSet<Decoration, SeqType> [polymake wiki]

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====== BigObject PartiallyOrderedSet<Decoration, SeqType> ======
//from application [[..:graph|graph]]//\\
\\
 A PartiallyOrderedSet is a poset where join and meet exist for any two elements. It is realized as a directed graph. Some implementations currently restricted to ranked posets (will be fixed soon™).

  ? Type Parameters:
  :: ''Decoration'': additional data associated with each node.  Should be derived from ''[[..:graph#BasicDecoration |BasicDecoration]]''.
  :: ''SeqType'': tag describing the node ordering, should be ''[[..:graph#Sequential |Sequential]]'' or ''[[..:graph#Nonsequential |Nonsequential]]''.
  ? derived from:
  : ''[[..:graph:Graph |Graph]]''
  ? Specializations:
  :: ''PartiallyOrderedSet::Basic'':  A ''[[..:graph:PartiallyOrderedSet |PartiallyOrderedSet]]'' with a ''[[..:graph#BasicDecoration |BasicDecoration]]'', which corresponds to the legacy HasseDiagram type
===== Properties =====

==== Combinatorics ====
 These properties capture combinatorial information of the object.  Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.
----
{{anchor:bottom_node:}}
  ?  **''BOTTOM_NODE''**
  :: The index of the bottom node
    ? Type:
    :''[[..:common#Int |Int]]''
    ? Example:
    :: The following prints the bottom node of the face lattice of the 2-simplex (triangle):
    :: <code perl> > print simplex(2)->HASSE_DIAGRAM->BOTTOM_NODE;
 0
</code>


----
{{anchor:comparability_graph:}}
  ?  **''COMPARABILITY_GRAPH''**
  :: An edge signals the comparability among poset elements (without top and bottom). Index shift by -1 since bottom and top are missing. This is required per specification of the GraphAdjacency class
    ? Type:
    :''[[..:common#GraphAdjacency |GraphAdjacency]]<[[..:common#Undirected |Undirected]]>''


----
{{anchor:decoration:}}
  ?  **''DECORATION''**
  :: This is the data associated to each node. The prototype for this is ''[[..:graph#BasicDecoration |BasicDecoration]]'', which consists of properties face and rank.
    ? Type:
    :''[[..:common#NodeMap |NodeMap]]<[[..:common#Directed |Directed]],Decoration>''
    ? Example:
    :: The following prints this property of the face lattice of the 2-simplex (triangle):
    :: <code perl> > print simplex(2)->HASSE_DIAGRAM->DECORATION;
 ({} 0)
 ({0} 1)
 ({1} 1)
 ({2} 1)
 ({1 2} 2)
 ({0 2} 2)
 ({0 1} 2)
 ({0 1 2} 3)
</code>


----
{{anchor:dims:}}
  ?  **''DIMS''**
  :: Kept only for backwards compatibility. Basically encodes the ''[[..:graph:PartiallyOrderedSet#INVERSE_RANK_MAP |INVERSE_RANK_MAP]]'' in FaceLattice objects prior to 3.0.7
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Int |Int]]>''


----
{{anchor:faces:}}
  ?  **''FACES''**
  :: The face of each node, realized as a NodeMap. This property is kept for two reasons: As a convenient way to access only the face part of the decoration (in this case the property is temporary) and for reasons of backwards compatibility.
    ? Type:
    :''[[..:common#NodeMap |NodeMap]]<[[..:common#Directed |Directed]],[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''
    ? Example:
    :: The following prints the faces of the face lattice of the 2-simplex (triangle):
    :: <code perl> > print simplex(2)->HASSE_DIAGRAM->FACES;
 {}
 {0}
 {1}
 {2}
 {1 2}
 {0 2}
 {0 1}
 {0 1 2}
</code>


----
{{anchor:inverse_rank_map:}}
  ?  **''INVERSE_RANK_MAP''**
  :: This property provides an efficient way to enumerate all nodes of a given rank. Internally these are realized differently, depending on whether the PartiallyOrderedSet is ''[[..:graph#Sequential |Sequential]]'' or ''[[..:graph#Nonsequential |Nonsequential]]''. Both provide the same user methods though. Notice that this function is necessary for technical reasons (for any PartiallyOrderedSet, even if it has maximal chains of various lengths). In fact, a "rank map" for our purpose is any assignment of natural numbers to the elements of a poset such that the (total) ordering of the numbers refines the (partial) ordering of the corresponding elements.
    ? Type:
    :''[[..:graph#InverseRankMap |InverseRankMap]]<SeqType>''
    ? Example:
    :: The following prints this property of the face lattice of the 2-simplex (triangle), where the tuples represent the ranges of nodes belonging to a specific rank:
    :: <code perl> > print simplex(2)->HASSE_DIAGRAM->INVERSE_RANK_MAP;
 {(0 (0 0)) (1 (1 3)) (2 (4 6)) (3 (7 7))}
</code>


----
{{anchor:maximal_anti_chains:}}
  ?  **''MAXIMAL_ANTI_CHAINS''**
  :: Maximal anti-chains
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:maximal_chains:}}
  ?  **''MAXIMAL_CHAINS''**
  :: Maximal chains
    ? Type:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''


----
{{anchor:top_node:}}
  ?  **''TOP_NODE''**
  :: The index of the top node
    ? Type:
    :''[[..:common#Int |Int]]''
    ? Example:
    :: The following prints the top node of the face lattice of the 2-simplex (triangle):
    :: <code perl> > print simplex(2)->HASSE_DIAGRAM->TOP_NODE;
 7
</code>


----
===== Methods =====

==== Combinatorics ====
 These methods capture combinatorial information of the object.  Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.
----
{{anchor:dual_faces:}}
  ?  **''dual_faces()''**
  ::
    ? Returns:
    :''[[..:common#Array |Array]]<[[..:common#Set |Set]]<[[..:common#Int |Int]]%%>>%%''
    ? Example:
    :: The following prints the dual faces of the face lattice of the 2-simplex (triangle):
    :: <code perl> > print simplex(2)->HASSE_DIAGRAM->dual_faces();
 {0 1 2}
 {1 2}
 {0 2}
 {0 1}
 {0}
 {1}
 {2}
 {}
</code>


----
{{anchor:nodes_of_rank:}}
  ?  **''nodes_of_rank([[..:common#Int |Int]] r)''**
  ::
    ? Parameters:
    :: ''[[..:common#Int |Int]]'' ''r''
    ? Returns:
    :''[[..:common#List |List]]<[[..:common#Int |Int]]>''
    ? Example:
    :: The following prints the nodes of rank 1 of the face lattice of the 2-simplex (triangle):
    :: <code perl> > print simplex(2)->HASSE_DIAGRAM->nodes_of_rank(1);
 {1 2 3}
</code>


----
{{anchor:nodes_of_rank_range:}}
  ?  **''nodes_of_rank_range([[..:common#Int |Int]] r1, [[..:common#Int |Int]] r2)''**
  ::
    ? Parameters:
    :: ''[[..:common#Int |Int]]'' ''r1''
    :: ''[[..:common#Int |Int]]'' ''r2''
    ? Returns:
    :''[[..:common#List |List]]<[[..:common#Int |Int]]>''
    ? Example:
    :: The following prints the nodes with rank between 1 and 2 of the face lattice of the 2-simplex (triangle):
    :: <code perl> > print simplex(2)->HASSE_DIAGRAM->nodes_of_rank_range(1,2);
 {1 2 3 4 5 6}
</code>


----
{{anchor:rank:}}
  ?  **''rank()''**
  ::
    ? Returns:
    :''[[..:common#Int |Int]]''
    ? Example:
    :: The following prints the rank of the top node of the face lattice of the 2-simplex (triangle):
    :: <code perl> > print simplex(2)->HASSE_DIAGRAM->rank();
 3
</code>


----

==== Visualization ====
 These methods are for visualization.
----
{{anchor:visual:}}
  ?  **''VISUAL()''**
  :: Visualize the partially ordered set.
    ? Options:
    : 
    :: ''[[..:common#Int |Int]]'' ''seed'': random seed value for the node placement
    : option list ''[[..:graph#Visual_Lattice_decorations |Visual::Lattice::decorations]]''
    ? Returns:
    :''[[..:graph:Visual_Lattice |Visual::Lattice]]''
    ? Example:
    :: The following visualizes the face lattice of the 2-simplex (triangle) with default settings:
    :: <code perl> > simplex(2)->HASSE_DIAGRAM->VISUAL;
</code>
    ::  The following shows some modified visualization style of the same lattice:
    :: <code perl> > simplex(2)->HASSE_DIAGRAM->VISUAL(NodeColor=>"green",EdgeThickness=>2,EdgeColor=>"purple");
</code>


----
{{anchor:visual_dual:}}
  ?  **''VISUAL_DUAL()''**
  :: Visualize the dual partially ordered set This only produces meaningful results for lattice where the codimension one nodes generate the lattice under intersection.
    ? Options:
    : 
    :: ''[[..:common#Int |Int]]'' ''seed'': random seed value for the node placement
    : option list ''[[..:graph#Visual_Lattice_decorations |Visual::Lattice::decorations]]''
    ? Returns:
    :''[[..:graph:Visual_Lattice |Visual::Lattice]]''
    ? Example:
    :: The following visualizes the dual face lattice of the 2-simplex (triangle) with default settings:
    :: <code perl> > simplex(2)->HASSE_DIAGRAM->VISUAL_DUAL;
</code>
    ::  The following shows some modified visualization style of the same lattice:
    :: <code perl> > simplex(2)->HASSE_DIAGRAM->VISUAL_DUAL(NodeColor=>"green",EdgeThickness=>2,EdgeColor=>"purple");
</code>


----