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Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2
BigObject Lattice<Decoration, SeqType>
from application graph
A Lattice is a poset where join and meet exist for any two elements. It is realized as a directed graph.
 Type Parameters:
Decoration
: additional data associated with each node. Should be derived fromBasicDecoration
.SeqType
: tag describing the node ordering, should beSequential
orNonsequential
. derived from:
 Specializations:
Lattice::BasicLattice
: ALattice
with aBasicDecoration
, 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.

BOTTOM_NODE
The index of the bottom node
 Type:
 Example:
The following prints the bottom node of the face lattice of the 2simplex (triangle):
> print simplex(2)>HASSE_DIAGRAM>BOTTOM_NODE; 0

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:

DECORATION
This is the data associated to each node. The prototype for this is
BasicDecoration
, which consists of properties face and rank. Type:
 Example:
The following prints this property of the face lattice of the 2simplex (triangle):
> 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)

DIMS
Kept only for backwards compatibility. Basically encodes the
INVERSE_RANK_MAP
in FaceLattice objects prior to 3.0.7 Type:

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:
 Example:
The following prints the faces of the face lattice of the 2simplex (triangle):
> print simplex(2)>HASSE_DIAGRAM>FACES; {} {0} {1} {2} {1 2} {0 2} {0 1} {0 1 2}

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 Lattice is
Sequential
orNonsequential
. Both provide the same user methods though. Type:
InverseRankMap<SeqType>
 Example:
The following prints this property of the face lattice of the 2simplex (triangle), where the tuples represent the ranges of nodes belonging to a specific rank:
> print simplex(2)>HASSE_DIAGRAM>INVERSE_RANK_MAP; {(0 (0 0)) (1 (1 3)) (2 (4 6)) (3 (7 7))}

MAXIMAL_ANTI_CHAINS
Maximal antichains (without top and bottom).
 Type:

MAXIMAL_CHAINS
Maximal chains (without top and bottom).
 Type:

TOP_NODE
The index of the top node
 Type:
 Example:
The following prints the top node of the face lattice of the 2simplex (triangle):
> print simplex(2)>HASSE_DIAGRAM>TOP_NODE; 7
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.

dual_faces()
 Returns:
 Example:
The following prints the dual faces of the face lattice of the 2simplex (triangle):
> print simplex(2)>HASSE_DIAGRAM>dual_faces(); {0 1 2} {1 2} {0 2} {0 1} {0} {1} {2} {}

nodes_of_rank(Int r)

rank()
 Returns:
 Example:
The following prints the rank of the top node of the face lattice of the 2simplex (triangle):
> print simplex(2)>HASSE_DIAGRAM>rank(); 3
Visualization
These methods are for visualization.

VISUAL()
Visualize the Lattice.
 Options:
Int
seed
: random seed value for the node placement option list
Visual::Lattice::decorations
 Returns:
 Example:
The following visualizes the face lattice of the 2simplex (triangle) with default settings:
> simplex(2)>HASSE_DIAGRAM>VISUAL;
The following shows some modified visualization style of the same lattice:
> simplex(2)>HASSE_DIAGRAM>VISUAL(NodeColor=>"green",EdgeThickness=>2,EdgeColor=>"purple");

VISUAL_DUAL()
Visualize the dual Lattice. This only produces meaningful results for lattice where the codimension one nodes generate the lattice under intersection.
 Options:
Int
seed
: random seed value for the node placement option list
Visual::Lattice::decorations
 Returns:
 Example:
The following visualizes the dual face lattice of the 2simplex (triangle) with default settings:
> simplex(2)>HASSE_DIAGRAM>VISUAL_DUAL;
The following shows some modified visualization style of the same lattice:
> simplex(2)>HASSE_DIAGRAM>VISUAL_DUAL(NodeColor=>"green",EdgeThickness=>2,EdgeColor=>"purple");