DrawSimplePosets

Create and draw Hasse diagrams. This is a companion to DrawSimpleGraphs.

NOTE: This is a work in progress; more to come.

Overview

A Hasse diagram is a drawing of a partially ordered set. Given a poset P, we construct a drawing of P like this:

H = HasseDiagram(P)

At this point, H does not have an embedding of the elements of P.

The function setxy!(H, xy) is used to assign coordinates to the elements of the poset. Here xy is a dictionary mapping elements of P (the poset used to create H) to points in the plane represented as a two-element vector. That is, if P is a SimplePoset whose elements have type T, then xy is of type Dict{T, Vector{Float64}}.

Standard call: H=HasseDiagram(P); setxy!(H, method(P))

If setxy! is invoked as setxy!(H) then a default embedding is assigned to H (using the basic_embedding function).

Finally, calling draw(H) will draw the poset P on the screen using the embedding associated with H.

For convenience, if H does not have an embedding, one will be automatically assigned to it using the basic_embedding function.

Finally, if draw is applied to a poset P, then draw(P) is equivalent to draw(HasseDiagram(P)).

julia> using SimplePosets, DrawSimplePosets

julia> P = Divisors(120)
SimplePoset{Int64} (16 elements)

julia> draw(P)

results in this image:

The draw function takes an optional second argument, clear_first, which is true by default. Use draw(H,false) to draw a Hasse diagram without first clearing the screen. Alternatively, draw!(H) has the same effect.

Basic Functions

Construction

If P is a SimplePoset, then HasseDiagram(P) creates a Hasse diagram of P.

Specifying coordinates

Given a Hasse diagram H of a poset P the function setxy! is used to assign coordinates to the elements of P. The function setxy! is applied to the Hasse diagram like this:

setxy!(H, xy)

where xy is a dictionary mapping elements of the poset to coordinates in the plane. That is, if P is a SimplePoset whose elements have type T, then xy is of type Dict{T, Vector{Float64}}.

Calling setxy!(H) embues H with coordinates created by basic_embedding.

Changing just the x- or y-coordinates.

The functions setx! and sety! can be applied to Hasse diagrams to modify the x- or y-coordinates (and leaving the other coordinates unchanged).

Let zz be a dictionary mapping (a subset of) the vertices of a poset represented by the Hasse diagram H to real numbers.

• setx!(H,zz) changes the x-coordinate of v to zz[v] (leaving v's y-coordinate unchanged).
• sety!(H,zz) changes the y-coordinate of v to zz[v] (leaving v's x-coordinate unchanged).

Getting coordinates

Use getxy(H) to return (a copy of) the dictionary mapping the elements of the poset represented by H to coordinates (values of type Vector{Float64}).

Shifting and scaling

Hasse diagrams can be shifted and scaled using these functions:

• shift!(H,dx,dy) modifies H by adding dx to all x-coordinates and dy to all y-coordinates.
• scale!(H,mx,my) modifies H by multiplying mx to all x-coordinates and my to all y-coordinates.

The function bounds(H) returns a 4-tuple (xmin,ymin,xmax,ymax) that gives the boundary of the Hasse diagram drawing. Note that the circles representing the elements of the poset may extend beyond the boundary.

Embedding Functions

This section will list the available embedding functions. At present, there is only one: basic_embedding. I hope we have others "eventually".

In general, an embedding function f takes a SimplePoset as its argument and returns an xy embedding for that poset.

basic_embedding

The function basic_embedding creates xy-coordinates for a poset, P as follows.

The minimal elements of P are all assigned y-coordinate 1. The x-coordinates are chosen so these elements are evenly spaced, distance 1 apart, and overall centered at 0. For example, if there are three minimal elements, their coordinates will be [-1, 1], [0, 1], and [1,1]. If there are four minimal elements, their x-coordinates will be -1.5, -0.5, 0.5, and 1.5.

Next, the minimal elements of P are discarded [no change is made to the poset] and the minimal elements of the remainder are assigned y-coordinate 2. The x-coordinates of these minimals are assigned as previously described.

Now these minimal elements are discarded, the minimals of the remainder are assigned y-coordinate 3, and so forth.

Graph methods that may be used on HasseDiagrams

The following functions defined for UndirectedGraphs may be applied to Hasse diagrams:

• draw_labels
• get_vertex_size
• set_vertex_size
• get_vertex_color
• set_vertex_color
• get_line_color
• set_line_color

DrawSimplePosets versus DrawSimpleGraphs

These two modules have differing design philosophies.

Drawings of undirected graphs using SimpleGraphs and DrawSimpleGraphs are attached to the graphs themselves. That is, the xy-embedding of an UndirectedGraph is part of the data structure of that graph.

For partially ordered sets we have taken a different approach. A SimplePoset object contains no xy-embedding information. Rather, a HasseDiagram created from that poset has the embedding. It is possible to create two (or more) Hasse diagrams of the same poset, each with its own embedding.

Under the Hood

A HasseDiagram object H has two fields:

• P which is the poset from which H was created.
• G which is an UndirectedGraph whose vertices are the elements of P and in which there is an edge from u to v if either u is a cover for v or v is a cover for u. These are the edgeds that are drawn in the diagram.

After H = HasseDiagram(P) the field H.P is the same exact object as P. The field H.G is created from P by finding the covering digraph and then ignoring directions. Subsequent changes to P will affect H.P but not H.G.