This folder contains extra code that may be interesting/useful in working with partially ordered sets.

Some of these files are sequestered here because they load other packages. Some might be of marginal value. Some might be incorporated into Posets in the future.

Please see the HasseDiagram package that supercedes the pplot.jl and posetplot.jl files that were in this directory.

Function provided:

• containment_order(list): Given a list of distinct objects for which issubseteq is defined, create a poset p in which i<j exactly when list[i] ⊆ list[j].

This file provides the function poset_converter that may be use to convert a poset defined in this Posets module (called Poset) to a poset (called SimplePoset) in the SimplePosets module. A simple call to poset_convert(p) converts p from one type to the other.

Note: The functions defined in the Poset and SimplePoset have a lot of the same names. This can cause conflicts. It may be necessary to invoke a function f either as Posets.f or SimplePosets.f to deal with the clash.

Function provided:

• divisors_poset(n): Create a poset whose elements correspond to the divisors of the (positive) integer n. In this poset we have a < b provided the a-th divisor of n is a factor of the b-th divisor.
• This can also be invoked as divisor_poset(list) where list is a list (Vector) of distinct positive integers to be ordered by divisibility.

Functions provided:

• semiorder(xs) creates a semiorder. Here xs is a list of n real numbers. The result is a poset with n elements in which i<j when x[i] ≤ x[j]-1. More generally, use semiorder(xs,t) in which case i<j when x[i] ≤ x[j]-t. Setting t=0 gives a total order (if the values in xs are distinct). If t is negative, errors may be thrown.

• interval_order(JJ) creates an interval order. Here JJ is a vector of ClosedIntervals. In this poset we have a < b provided JJ[a] lies entirely to the left of JJ[b].

• random_interval_order(n) creates a random interval order with n elements. This is done by creating n random intervals and invoking interval_order.

The function partition_lattice(n) returns a pair (p, tab).

• p is the poset containing all partitions of the set {1,2,...,n} ordered by refinement. The least element of this poset is {{1},{2},...,{n}} and the largest element of this poset is {{1,2,...,n}}.
• tab is table containing the partitions of {1,2,...,n} so that element a of p corresponds to the parition tab[a].

The function partition_lattice_demo prints out a maximal chain in a partition lattice.

julia> using ShowSet

julia> partition_lattice_demo(5)
{{1},{2},{4},{3},{5}} < {{1,5},{2},{4},{3}} < {{4},{3},{1,2,5}} < {{1,2,4,5},{3}} < {{1,2,3,4,5}}