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 from BasicDecoration.

SeqType: tag describing the node ordering, should be Sequential or Nonsequential.

derived from:

Graph

Specializations:

Lattice::BasicLattice: A Lattice with a BasicDecoration, which corresponds to the legacy HasseDiagram type

These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

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BOTTOM_NODE

The index of the bottom node

Type:

Int

Example:

The following prints the bottom node of the face lattice of the 2-simplex (triangle):

 > print simplex(2)->HASSE_DIAGRAM->BOTTOM_NODE;
 0

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DECORATION

This is the data associated to each node. The prototype for this is BasicDecoration, which consists of properties face and rank.

Type:

NodeMap<Directed,Decoration>

Example:

The following prints this property of the face lattice of the 2-simplex (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)

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DIMS

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

Type:

Array<Int>

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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:

NodeMap<Directed,Set<Int>>

Example:

The following prints the faces of the face lattice of the 2-simplex (triangle):

 > print simplex(2)->HASSE_DIAGRAM->FACES;
 {}
 {0}
 {1}
 {2}
 {1 2}
 {0 2}
 {0 1}
 {0 1 2}

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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 or Nonsequential. Both provide the same user methods though.

Type:

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:

 > print simplex(2)->HASSE_DIAGRAM->INVERSE_RANK_MAP;
 {(0 (0 0)) (1 (1 3)) (2 (4 6)) (3 (7 7))}

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TOP_NODE

The index of the top node

Type:

Int

Example:

The following prints the top node of the face lattice of the 2-simplex (triangle):

 > print simplex(2)->HASSE_DIAGRAM->TOP_NODE;
 7

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These methods capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

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dual_faces()

Returns:

Array<Set<Int>>

Example:

The following prints the dual faces of the face lattice of the 2-simplex (triangle):

 > print simplex(2)->HASSE_DIAGRAM->dual_faces();
 {0 1 2}
 {1 2}
 {0 2}
 {0 1}
 {0}
 {1}
 {2}
 {}

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nodes_of_rank(Int r)

Parameters:

Int r

Returns:

List<Int>

Example:

The following prints the nodes of rank 1 of the face lattice of the 2-simplex (triangle):

 > print simplex(2)->HASSE_DIAGRAM->nodes_of_rank(1);
 {1 2 3}

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nodes_of_rank_range(Int r1, Int r2)

Parameters:

Int r1

Int r2

Returns:

List<Int>

Example:

The following prints the nodes with rank between 1 and 2 of the face lattice of the 2-simplex (triangle):

 > print simplex(2)->HASSE_DIAGRAM->nodes_of_rank_range(1,2);
 {1 2 3 4 5 6}

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rank()

Returns:

Int

Example:

The following prints the rank of the top node of the face lattice of the 2-simplex (triangle):

 > print simplex(2)->HASSE_DIAGRAM->rank();
 3

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These methods are for visualization.

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VISUAL()

Visualize the Lattice.

Options:

Int seed: random seed value for the node placement

option list Visual::Lattice::decorations

Returns:

Visual::Lattice

Example:

The following visualizes the face lattice of the 2-simplex (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");

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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:

Visual::Lattice

Example:

The following visualizes the dual face lattice of the 2-simplex (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");

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